![Antoine Henrot
1 Introduction
1.1 General introduction
This collective book has the ambition to give an overview of recent results in spectral
geometry and its links with shape optimization. The questions we are interested in
are:
– Does there exist a set which minimizes (or maximizes) the k-th eigenvalue of a
given elliptic operator (mainly the Laplacian or the Schrödinger operator) with
given boundary conditions (mainly Dirichlet, Robin or Steklov boundary condi-
tions in this book) among sets of given volume?
– If existence is proved, what can be said about the regularity of the optimal set?
– Can one compute the optimal set? Can one give some geometric properties of it?
– Is it possible to prove some stability results for the well-known isoperimetric in-
equalities involving the first or the second eigenvalue, namely some quantitative
isoperimetric inequalities for eigenvalues?
– More generally, can one write some universal inequalities related to the eigenval-
ues of elliptic operators?
– For some more specific shapes like triangles, can one prove more precise bounds
and observe some patterns of the eigenfunctions?
A first survey on these kinds of extremum problems appeared in 2006 in the book
"Extremum Problems for eigenvalues of elliptic operators", see [505]. It turns out that
much progress has been made in the last ten years which leads us to think that an
update and an extension would be valuable for a wide community who is interested
in spectral theory and its links with geometry, calculus of variations, free boundary
problems and shape optimization.
1.2 Content of the book
We now describe in more detail the content of each chapter. For most of the nota-
tion used here and in this book, we refer to the end of this Introduction chapter (Sec-
tion 1.4).
Antoine Henrot: Institut Élie Cartan, UMR 7502, Université de Lorraine, CNRS BP 70239, 54506
Vandoeuvre-lès-Nancy, France, E-mail: antoine.henrot@univ-lorraine.fr
© 2017 Antoine Henrot
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-15-320.jpg)
![2 | Antoine Henrot
Existence results by D. Bucur
In Chapter 2, existence of a minimizer for a problem like
min{λk(Ω), Ω ⊂ Rd
, |Ω| = c} (1.1)
or more generally like
min{F λ1(Ω), . . . λk(Ω)
, Ω ⊂ Rd
, |Ω| = c} (1.2)
is studied. It is necessary to enlarge the class of admissible sets to quasi-open sets
which is actually the largest, and more natural, class for which eigenvalues of the
Dirichlet-Laplacian are well defined. A famous existence result due to G. Buttazzo and
G. Dal Maso in [238] was previously available but with the supplementary assumption
that the sets have to lie in some fixed bounded box D. A lot of effort has been done to
remove this assumption and to get a general existence result. These efforts eventually
led to the papers by D. Bucur [206] and A. Pratelli-D. Mazzoleni [700] where this open
problem of existence was solved. These two papers use a completely different strategy.
In [206], the notion of shape subsolution for the torsion energy was introduced, and it
was proved that every such subsolution has to be bounded and has finite perimeter.
A second argument, showed that minimizers for (1.1) are shape subsolutions, so they
are bounded. The author finished the proof by using a concentration-compactness ar-
gument, like in [219]. The approach of [700] is different: a surgery result proved that
some parts (like long and tiny tentacles), can be cut out from every set such that, af-
ter small modifications and rescaling, the new set has a diameter uniformly bounded
and its first k eigenvalues are smaller. In this way, the existence problem in Rd
can be
reduced to the local case of Buttazzo and Dal Maso. In Chapter 2, the main ideas of
the proof of the global existence result are provided, using a combination of the two
methods: shape subsolutions and surgery. Moreover, it is proved that the optimal set
is bounded and has finite perimeter.
Regularity of optimal spectral domains by J. Lamboley and M. Pierre
Once existence has been proved (for example for Problem (1.1)), an important and dif-
ficult issue is to study the regularity of the optimal set Ω*
. As we have seen, the exis-
tence theorem provides a solution which is only quasi-open and this is a very weak
regularity. Now it seems reasonable to expect much more regularity such as Lipschitz
or C2
boundary, even analytic (at least in two dimensions). We would like to have this
kind of regularity to be able to write optimality conditions thanks to the shape deriva-
tive, see [510, chapter 5]. For computing such shape derivatives, a minimal regularity
is required (e.g. C2
regularity if we want to consider the trace of the gradient on the
boundary which occurs in the classical Hadamard’s formula). It turns out that the](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-16-320.jpg)
![Introduction | 3
main difficulty in this regularity issue is the first step: to be able to reach weak but
consistent regularity, for example that the boundary of the optimal domain is locally
a graph. Once this is done, it is often possible to use powerful tools developed in the
theory of regularity for free boundary problems (e.g. by L. Caffarelli and co-authors)
to reach the desired properties. Chapter 3 is devoted to these questions and presents
the known results. The authors also point out many open problems which remain to
be solved. In a first section, they consider the problem
min{λ1(Ω), Ω ⊂ D, |Ω| = c} (1.3)
where D is a given bounded domain. Of course, if D is large enough (or c small enough)
in order that it contains the ball of volume c, this one is the solution and there is noth-
ing to prove. Otherwise, the regularity of the optimal domain is studied in detail and is
well understood. The fact that the state function (the first eigenfunction u1) is positive
plays an important role in this analysis. Actually, the first step is to study the global
regularity of u1 naturally extended by zero outside Ω*
. For that purpose, the authors
show that Problem (1.3) is actually equivalent to a penalized version
min{λ1(Ω) + µ[|Ω| − c]+
, Ω ⊂ D} (1.4)
(where [x]+
denotes the positive part of x) for µ large enough. It turns out that it is
much more convenient to work with this penalized version, in particular it is easier to
perform variations and exploit the minimality of the domain which is used to prove
that the eigenfunction is globally Lipschitz continuous. This idea of the penalized ver-
sion works as well for any eigenvalue λk but without a box constraint: Problem (1.1) is
equivalent to
min{λk(Ω) + µ|Ω|, Ω ⊂ Rd
} (1.5)
for a particular value of µ. Knowing that the state function is Lipschitz continuous is
a first (important) step in the study of the regularity of the boundary of the optimal
set, but obviously not sufficient. The next step is to prove that the gradient of the state
function does not degenerate at the boundary, in order to be able to use some im-
plicit function theorem to deduce regularity of the boundary itself. This is done in this
chapter for λ1 where analyticity of the boundary is proved in dimension d = 2, while
regularity of the reduced boundary is proved in higher dimension. Then, for Problem
(1.3) the regularity of ∂Ω*
up to the boundary of the box D is studied. At last, some
problems with a convexity constraint are considered.
The Robin problem by D. Bucur, P. Freitas and J. Kennedy
While the two previous chapters deal with Dirichlet boundary conditions, the next
one deals with Robin boundary conditions: ∂u
∂n + αu = 0 on ∂Ω. This is a very impor-
tant case since it can be seen as a generalization (or interpolation) of the Dirichlet and](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-17-320.jpg)
![Introduction | 7
which were proved for (planar) domains by Payne, Pólya and Weinberger in [746] and
which may be considered as the starting point of that study. Other more general re-
sults have been obtained by Hile and Protter on the one hand and Yang on the other,
and Chapter 8 shows in a very clear way all these results and the hierarchy between
them. Let me also mention that the first inequality in (1.7) has been later improved
as λ2/λ1 ≤ j2
d/2,1/j2
d/2−1,1 by the author and R. Benguria in [62], [64] giving here the
sharp inequality. Chapter 8 presents in an unified way these universal inequalities by
choosing the more general approach. All the proofs are given and the author often
chooses to present simpler proofs or more general results than the original ones. In
particular, several statements are given with the Laplacian replaced by a Schrödinger
operator with both a scalar and vectorial potential. The authors considers whether it
is possible to deduce from the previous universal inequalities some explicit bounds
for a given eigenvalue, say λm+1. Moreover, we would like these bounds to be in good
accordance with Weyl’s law. Several interesting results in that direction are presented.
This chapter aims to put in perspective all of this material, by giving historical markers
and references.
Spectral optimization problems for Schrödinger operators by G. Buttazzo and B.
Velichkov
In this chapter Schrödinger operators of the form −∆+V(x) are considered, with Dirich-
let boundary conditions on a bounded open set D ⊂ Rd
. The question is now to find
optimal potentials for some suitable optimization criteria. In general, the optimiza-
tion problems studied here can be written as min{F(V), V ∈ V} where F is a suitable
cost functional and V is a suitable class of admissible potentials. The case of spectral
functionals min{Φ(λ(V)), V ∈ V}, where λ(V) is the spectrum of the Schrödinger op-
erator, which will be assumed to be discrete, are also considered. Here, the interest is
not only in potentials V which are bounded functions, but also in their natural exten-
sion: the capacitary measures which are nonnegative Borel measures on D, possibly
taking the value +∞ and vanishing on all sets of capacity zero. The class of capaci-
tary measures is very large and contains both the cases of standard potentials V(x),
in which µ = Vdx, as well as the case of classical domains, where we set µ = +∞DΩ
which is the measure defined by
µ(E) =
(
0 if cap(Ω E) = 0
+∞ if cap(Ω E) 0.
(1.8)
In that sense, this can be seen as a unified presentation of these two classical prob-
lems. The authors are mainly interested in existence theorems. They first prove a very
general existence result in the class of capacitary measures. Then they consider the
more specific case of integrable potentials: V ∈ Lp
(D) for which other general exis-
tence theorems are proved. Some examples where the optimal potential can be ex-](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-21-320.jpg)
![Introduction | 9
Numerical results for spectral optimization problems by P. Antunes and E. Oudet
This chapter is devoted to numerical methods which have been introduced to solve the
previous problems. In the first two sections two of these approaches which have been
successful in recent years on spectral problems are explained. The first one consists
in introducing some global optimization tools to provide a good initial guess of the
optimal profile. This step does not require any topological information on the set but
is restricted to a small class of shapes. Then the method of fundamental solutions (to
compute the eigenvalues) is described. It allows, in a second stage, the identification
and precise evaluation of shapes which are locally optimal. A constant preoccupation
is to decrease the complexity of the optimization problem by introducing a reduction
of the number of parameters which still allows a precise computation of the cost func-
tion. For example, the parametrization of the boundaries of the open sets as level set
functions, for example level sets of truncated Fourier series can be very efficient. The
chapter ends with a presentation of the best domains obtained numerically for both
Dirichlet and Neumann eigenvalues λk and µk (for k = 1 to 10 or 15) and some con-
jectures inspired by these numerical results.
1.3 Balls and union of balls
One of the most important topics discussed in this book is the determination of which
domain minimizes or maximizes a given eigenvalue. For low eigenvalues, actually the
two first eigenvalues, and for most boundary conditions, the optimal domains are
known and it turns out that they are the same: the ball for the first eigenvalue and
the union of two identical balls for the second. These results are recalled in different
chapters of this book, but let us sum up it here.
First eigenvalue-Dirichlet The ball minimizes λ1(Ω) among sets of given volume
(Faber-Krahn inequality), see [377] and [603].
First (non-trivial) eigenvalue-Neumann The ball maximizes µ2(Ω) among sets of
given volume (Szegő-Weinberger inequality) , see [838] for Lipschitz simply con-
nected planar domains and [871] for the general case.
First eigenvalue-Robin (α 0) The ball minimizes λ1(Ω, α) among sets of given vol-
ume (Bossel-Daners inequality), see [169] for the two-dimensional case and [319]
for the general case.
First eigenvalue-Steklov The ball maximizes σ2(Ω) among sets of given volume
(Brock-Weinstock inequality), see [873] for the two-dimensional case and [192] for
the general case.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-23-320.jpg)
![10 | Antoine Henrot
Fig. 1.1. Left: the disk minimizes the first eigenvalue (Dirichlet or Robin) and maximizes the first non
trivial eigenvalue (Neumann or Steklov). Right: two disks minimizes the second eigenvalue (Dirichlet
or Robin) and maximizes the second non trivial eigenvalue (Neumann or Steklov)
Second eigenvalue-Dirichlet The union of two identical balls minimizes λ2(Ω)
among sets of given volume (Hong-Krahn-Szego inequality), see [604], [534].
Second (non-trivial) eigenvalue-Neumann The union of two identical disks maxi-
mizes µ3(Ω) among simply connected bounded planar domains of given volume,
see [428].
Second eigenvalue-Robin (α 0) The union of two identical balls minimizes
λ1(Ω, α) among sets of given volume, see [586] and Theorem 4.36 in Chapter 4.
Second eigenvalue-Steklov The union of two identical disks maximizes σ3(Ω)
among simply connected bounded planar domains of given volume, see [513],
[430] and Chapter 5.
In view of the previous results, it is a natural question to ask whether there are other
eigenvalues for which balls or union of balls could be the optimal domain. For Dirich-
let eigenvalues, this question has been recently investigated in the PhD thesis of A.
Berger, see [137]. She proves that for d = 2, only λ1 and λ3 can be minimized by the
disk (it is still a conjecture for λ3). Moreover, only λ2 and λ4 can be minimized by union
of disks (it is still a conjecture for λ4).
Let us finish this section by giving the eigenvalues of the ball. In dimension 2,
the eigenvalues of the disk BR of radius R for the Laplacian with Dirichlet boundary
conditions and the corresponding eigenfunctions (not normalized) are given by
λ0,k =
j2
0,k
R2 , k ≥ 1,
u0,k(r, θ) = J0(j0,kr/R), k ≥ 1,
λn,k =
j2
n,k
R2 , n, k ≥ 1, double eigenvalue
un,k(r, θ) =
(
Jn(jn,kr/R) cos nθ
Jn(jn,kr/R) sin nθ
, n, k ≥ 1,
(1.10)
where jn,k is the k-th zero of the Bessel function Jn.
For the Laplacian with Neumann boundary conditions, the eigenvalues and eigen-](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-24-320.jpg)
![Introduction | 11
functions of the disk BR are:
µ0,k =
j′
0,k
2
R2 , k ≥ 1,
v0,k(r, θ) = J0(j′
0,kr/R), k ≥ 1,
µn,k =
j′
n,k
2
R2 , n, k ≥ 1, double eigenvalue
vn,k(r, θ) =
(
Jn(j′
n,kr/R) cos nθ
Jn(j′
n,kr/R) sin nθ
, n, k ≥ 1,
(1.11)
where j′
n,k is the k-th zero of J′
n (the derivative of the Bessel function Jn).
In dimension three, for the ball BR, the eigenvalues and eigenfunctions of the
Dirichlet-Laplacian are given by λn,k = j2
n+ 1
2 ,k
/R2
, n ∈ N, k ∈ N which is of multi-
plicity 2n + 1 and is associated to the eigenfunctions
vn,k(r, θ, ϕ) =
Jn+ 1
2
j
n+ 1
2
,k
R r
!
√
r
P0
n(cos θ),
Jn+ 1
2
j
n+ 1
2
,k
R r
!
√
r
P1
n(cos θ) cos ϕ,
Jn+ 1
2
j
n+ 1
2
,k
R r
!
√
r
P1
n(cos θ) sin ϕ,
.
.
.
Jn+ 1
2
j
n+ 1
2
,k
R r
!
√
r
Pn
n(cos θ) cos(nϕ),
Jn+ 1
2
j
n+ 1
2
,k
R r
!
√
r
Pn
n(cos θ) sin(nϕ)
(1.12)
where Pq
n denote the associated Legendre polynomial, see [4]. Similar formulae hold
for the Neumann eigenvalues and eigenfunctions where the eigenvalues are the
roots of some transcendental equations involving Bessel functions. In higher dimen-
sion d, the eigenvalues of the ball BR still involve the zeros of the Bessel functions
Jd/2−1, Jd/2, . . .. For example
λ1(BR) =
j2
d/2−1,1
R2
λ2(BR) = λ3(BR) = . . . = λN+1(BR) =
j2
d/2,1
R2
(1.13)
while the eigenfunctions combine Bessel functions for the radial part and spherical
harmonics for the angular part, see [292]
1.4 Notation
Ω is an open set (or a quasi-open set, see Chapter 2) in Rd
. We will denote by H1
(Ω)
the classical Sobolev space:](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-25-320.jpg)
![Dorin Bucur
2 Existence results
2.1 Setting the problem
In this chapter, we denote by d the dimension of the space. Let d ≥ 2 and Ω ⊆ Rd
be an
open set of finite measure. Then the spectrum of the Laplace operator with Dirichlet
boundary conditions of Ω consists only on eigenvalues which can be ordered (count-
ing the multiplicity)
0 λ1(Ω) ≤ λ2(Ω) ≤ · · · → +∞.
For every k ∈ N there exist non-zero functions u (eigenfunctions) that saisfy the equa-
tion (
−∆u = λk(Ω)u in Ω
u = 0 on ∂Ω.
If Ω is smooth, then the function u ∈ C2
(Ω)∩ C(Ω) satisfies the equation in a classical
sense. If Ω is just an open set without any regularity, then u satisfies the equation in
the following weak sense
u ∈ H1
0(Ω), ∀v ∈ H1
0(Ω),
ˆ
Ω
∇u∇vdx = λk(Ω)
ˆ
Ω
uvdx.
Let c 0, k ∈ N be given, and F : Rk
→ R. The generic spectral optimization
problems we discuss in this chapter is
min
n
F λ1(Ω), .., λk(Ω)
: Ω ⊂ Rd
, Ω open, |Ω| = c
o
. (2.1)
Some particular problems have been studied intensively in the last century. We
refer the reader to [505] for a recent survey of the topic. Here is a short list of results.
– The Faber-Krahn inequality asserts that the solution of
min
n
λ1(Ω) : Ω ⊂ Rd
, Ω open, |Ω| = c
o
(2.2)
is the ball of volume c.
– The solution of
min
n
λ2(Ω) : Ω ⊂ Rd
, Ω open, |Ω| = c
o
(2.3)
consists of two equal and disjoint balls of volume c
2 (Krahn-Szegő).
Dorin Bucur: Institut Universitaire de France, Laboratoire de Mathématiques, CNRS UMR
5127, Université de Savoie, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France, E-mail:
dorin.bucur@univ-savoie.fr
© 2017 Dorin Bucur
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-27-320.jpg)
![14 | Dorin Bucur
– Ashbaugh and Benguria proved in [64] that the solution of
max
n λ2(Ω)
λ1(Ω)
: Ω ⊂ Rd
, Ω open and of finite measure
o
(2.4)
is the ball.
An intriguing question is to find the solution of
min
n
λk(Ω) : Ω ⊂ Rd
, Ω open, |Ω| = c
o
(2.5)
for every k ∈ N. Unfortunately, starting with k ≥ 3 very few answers are available.
In two dimensions, for k = 3 it is conjectured that the minimizer is the disc, while in
dimension 3 it has been observed numerically by Oudet that the minimizer is not the
ball, cf Chapter 11.
Wolf and Keller proved that for k = 13, in R2
, the minimizer is not a union of discs,
but again Oudet [737] numerically observed that for k = 5 to 15 the minimizer is not
the disc. In [137], it was rigorously proved by Berger, that for any k ≥ 5 in R2
, the ball
cannot be minimizer.
Several computations were carried out ([42, 737]) providing evidence that the op-
timal shapes are close to those presented in Chapter 11.
At this point, when no analytical solutions of those problems can be expected, the
question is of a qualitative nature. One would like to prove that problem (2.1), or more
precisely problem (2.5), has a solution and to gather some information about it. Does
the optimal set have finite perimeter ? Is it bounded ? Is its boundary smooth ? Does it
have any symmetry ? Is it convex ? Is it the ball ?
Problem (2.1) may have or not a solution (for general functions F, the complete
answer is not known), but a negative answer may have at least three meanings:
– a solution of problem (2.1) does not exist (i.e. in the class of open sets), but there
exists a solution provided the family of open sets is enlarged to a class of Borel
subsets of Rd
where the eigenvalue problem is still well posed (i.e. the family of
quasi-open sets, see Section 2.2 below). This issue is very similar to a classical ex-
istence result, only that the solution is a quasi-open set. In fact the class of quasi-
open sets is the largest class of Borel subsets of Rd
, where the Dirichlet Laplacian
is well defined and inherits a strong maximum principle.
– a solution of problem (2.1) does not exist, even if the class of sets is enlarged, but
there exists a solution in a larger class of relaxed objects where the eigenvalue
problem is well posed. This class consists of positive Borel measures, absolutely
continuous with respect to capacity (see Section 2.2 below for definition and prop-
erties of capacity). Roughly speaking, those measures are limits of sequences of
open sets in some suitable sense, and account for the asymptotic behavior of the
oscillating boundaries in the sense of capacities (see Remark 2.5 below).
– a solution of problem (2.1) does not exist, in the sense that the infimum is not
attained by any geometrical object.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-28-320.jpg)
![2 Existence results | 15
In this chapter, we shall analyze the existence of a solution for problem (2.1), and
we shall prove that it indeed exists (in the enlarged class of quasi-open sets), provided
some assumptions are satisfied by the functional F. Of course, one expects to have
smooth open sets as minimizers, at least for problem (2.5), but this has not yet been
proved in general. The question of proving existence in the class of open sets is in
fact a regularity problem for which we refer the reader to Chapter 3. Some qualitative
properties will be proved in this chapter, e.g. the boundedness of the optimal sets and
the fact that they have a finite perimeter, since they play a crucial role in the existence
question.
2.2 The spectrum of the Dirichlet-Laplacian on open and
quasi-open sets
Since the existence question requires us to work in a class of sets larger than the class
of open sets, in this section we recall basic facts about capacity and quasi-open sets.
We also list some properties of the eigenvalues of the Dirichlet-Laplacian and of the
eigenfunctions.
Capacity and quasi-open sets. Let E ⊆ Rd
. The capacity of E is defined by
cap(E) = inf
n ˆ
|∇u|2
+ |u|2
dx, u ∈ UE
o
where UE is the class of all functions u ∈ H1
(Rd
) such that u ≥ 1 almost everywhere
(shortly a.e.) in an open neighborhood of E.
A property p(x) is said to hold quasi everywhere on E (shortly q.e. on E) if the set
of all points x ∈ E for which the property p(x) does not hold has capacity zero.
A set Ω ⊆ Rd
is called quasi-open if for every ϵ 0 there exists an open set Uϵ
such that Ω ∪ Uϵ is open and cap(Uϵ) ϵ. Clearly, every open set is quasi-open. A
function u : Rd
7→ R is said to be quasi-continuous if for all ϵ 0 there exists an open
set Uϵ with cap(Uϵ) ϵ such that u|Uc
ϵ
is continuous (see [472]).
Every function u ∈ H1
(Rd
) has a quasi-continuous representative, ũ, such that
ũ(x) = u(x) a.e. This representative is unique up to a set of zero capacity and can be
computed by
ũ(x) = lim
r→0
´
Br(x)
u(y)dy
|Br(x)|
, q.e. x ∈ Rd
.
The limit above exists quasi everywhere. In particular, the level set {ũ 0} is a quasi-
open set. From now on, every time we speak about the pointwise behavior of a Sobolev
function, we refer to a quasi-continuous representative.
The Sobolev spaces. If Ω ⊆ Rd
is an open set, the Sobolev space H1
0(Ω) is defined as
clH1(Rd)C∞
0 (Ω), the closure of the space of C∞
functions with compact support in Ω, in](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-29-320.jpg)
![16 | Dorin Bucur
the H1
-norm. For a quasi-open set Ω ⊆ Rd
, the Sobolev space H1
0(Ω) is defined as a
subspace of H1
(Rd
) by:
H1
0(Ω) = {u ∈ H1
(Rd
) : u = 0 q.e. on Rd
Ω}.
If Ω is open, the space H1
0(Ω) defined above coincides with the usual Sobolev space
(see [472]).
From this perspective, for every open or quasi-open set, H1
0(Ω) is a subspace of
H1
(Rd
), as long as every function of H1
0(Ω) is understood as being extended by 0 on
Rd
Ω.
If Ω ⊆ Rd
is a quasi-open set of finite measure (not necessarily bounded), the
injection H1
0(Ω) ,→ L2
(Ω) is compact.
The spectrum of the Dirichlet-Laplacian. For every quasi-open set of finite measure
Ω ⊆ Rd
, we introduce the resolvent operator RΩ : L2
(Rd
) → L2
(Rd
), by RΩ(f) = u,
where u solves the equation
(
−∆u = f in Ω
u ∈ H1
0(Ω)
in the weak sense
∀ϕ ∈ H1
0(Ω)
ˆ
Ω
∇u∇ϕdx =
ˆ
Ω
fϕdx. (2.6)
RΩ is a compact, self-adjoint, positive operator having a sequence of eigenvalues
converging to 0. The inverses of its eigenvalues are the eigenvalues of the Dirichlet-
Laplacian on Ω and are denoted (multiplicity being counted) by
0 λ1(Ω) ≤ λ2(Ω) ≤ · · · ≤ λk(Ω) ≤ · · · → +∞.
These values can be defined by the min-max formula
λk(Ω) = min
S∈Sk
max
u∈S{0}
´
Ω
|∇u|2
dx
´
Ω
u2dx
, (2.7)
where Sk stands for the family of all subspaces of dimension k in H1
0(Ω). A function u ∈
H1
0(Ω) for which equality holds, is called an eigenfunction and satisfies the equation
(
−∆u = λk(Ω)u in Ω
u = 0 on ∂Ω
in the weak sense
∀ϕ ∈ H1
0(Ω)
ˆ
Ω
∇u∇ϕdx = λk(Ω)
ˆ
Ω
uϕdx.
The system of L2
-normalized eigenfunctions is a Hilbert basis of H1
0(Ω).
We list below some properties of the eigenvalues. Let Ω, Ω1, Ω2 ⊆ Rd
be quasi-
open sets of finite measure.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-30-320.jpg)
![2 Existence results | 17
– (Rescaling) ∀t 0, ∀k ∈ N, λk(tΩ) = 1
t2 λk(Ω).
– (Spectrum of the union) If Ω1, Ω2 are disjoint, then the eigenvalues of Ω1 ∪ Ω2
are the union of the sets of eigenvalues of Ω1 and Ω2 with multiplicities being
counted.
– (Monotonicity) Assume that Ω1 ⊆ Ω2. Then
∀k ∈ N, λk(Ω2) ≤ λk(Ω1).
– (Control of the variation)
1
λk(Ω1)
−
1
λk(Ω2)
≤ kRΩ1
− RΩ2
kL(L2(Rd)).
Moreover, if Ω1 ⊆ Ω2 (see Bucur [206]), for every k ∈ N
1
λk(Ω1)
−
1
λk(Ω2)
≤ 4k2
e1/4π
λk(Ω2)d/2
(E(Ω1) − E(Ω2)). (2.8)
where
E(Ω) := min
u∈H1
0(Ω)
1
2
ˆ
Rd
|∇u|2
dx −
ˆ
Rd
udx. (2.9)
is the torsion energy. The unique function which minimizes E(Ω) is called the tor-
sion function and is denoted wΩ and satisfies in a weak sense
−∆wΩ = 1 in Ω, wΩ ∈ H1
0(Ω).
The torsion function plays a key role in understanding the behavior of the spec-
trum of the Dirichlet Laplacian for small geometric domain perturbations. Assume
that λk(Ω2) = K and Ω1 ⊆ Ω2 such that
E(Ω1) − E(Ω2) =
1
2
ˆ
Ω2
(wΩ2
− wΩ1
)dx ≤
1
8k2e1/4πKd/2+1
. (2.10)
Then we get a control on the magnitude of λk(Ω1). Precisely, if (2.10) holds, we get
λk(Ω1) ≤ 2λk(Ω2). (2.11)
As
ˆ
Ω2
(wΩ2
−wΩ1
)dx becomes smaller, the eigenvalues λk(Ω1) and λk(Ω2) become
closer.
Roughly speaking, this property asserts that the variation of the eigenvalues for
inner perturbations of a quasi-open set is controlled by the variation of the L1
-
norm of the torsion functions (see Section 2.5).
– (Control by the torsion function, see Van den Berg [860])
1
λ1(Ω)
≤ kwΩk∞ ≤
4 + 3d log 2
λ1(Ω)
. (2.12)](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-31-320.jpg)
![18 | Dorin Bucur
– (Ratio of eigenvalues) For all k ∈ N there exists a constant Mk, depending only on
k and the dimension d, such that (see for instance [57])
1 ≤
λk(Ω)
λ1(Ω)
≤ Mk. (2.13)
– (L∞
-bound of the eigenfunctions) If uk is an L2
-normalized eigenfunction of
λk(Ω), then
kukk∞ ≤ Cdλk(Ω)
d
4 . (2.14)
2.3 Existence results: bounded design region
Let D ⊆ Rd
be a bounded open set. In this section we shall prove an existence result
for a local version of problem (2.1), i.e.
min
F λ1(Ω), .., λk(Ω)
: Ω ⊂ D, Ω quasi-open, |Ω| = c (2.15)
Theorem 2.1. (Buttazzo-Dal Maso) Let F : Rk
→ R be non-decreasing in each vari-
able and lower semicontinuous. Then problem (2.15) has at least one solution.
The first proof of this theorem, given by Buttazzo and Dal Maso in [238], involved quite
technical results describing the so called relaxation phenomenon and covered a more
general situation than just a functional depending on the eigenvalues. We give below
a direct proof, which does not require the knowledge of the relaxed problem or prop-
erties of the weak gamma convergence (see Remark 2.6), since we are concerned only
with functionals depending on eigenvalues.
A series of remarks at the end of the proof will explain the necessity of the mono-
tonicity hypothesis on F and the boundedness of the design region D.
Proof. Assume that (Ωn)n is a minimizing sequence for problem (2.15) and that
u1
n, . . . , uk
n are L2
-normalized eigenfunctions corresponding to λ1(Ωn), . . . , λk(Ωn),
two by two orthogonal in L2
(D). We can assume that (λk(Ωn))n is a bounded sequence,
otherwise existence occurs trivially as a consequence of the monotonicity of F (from
some rank on, F will be constant on the sets Ωn). After extracting a subsequence we
can assume that (ui
n)n converges weakly in H1
0(D), strongly in L2
(D) and pointwise
a.e. to a function ui
∈ H1
0(D), for i = 1, . . . , k. We can also assume that ui
are quasi-
continuous, so that defining
Ω :=
k
[
i=1
{ui
6= 0},
we built a quasi-open set Ω ⊆ D such that ui
∈ H1
0(Ω). From the pointwise a.e. con-
vergence, we get
1{ui 6 = 0} ≤ lim inf
n→∞
1Ωn
a.e.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-32-320.jpg)
![2 Existence results | 19
so that 1Ω ≤ lim infn→∞ 1Ωn
a.e. This implies |Ω| ≤ c.
On the other hand
λi(Ω) ≤ lim inf
n→∞
λi(Ωn), ∀i = 1, . . . , k.
Indeed, this is a consequence of the definition of the eigenvalues. Let us denote Si =
span{u1
, . . . , ui
}. We have, for some (αj)i
j=1 ∈ Ri
λi(Ω) ≤ max
u∈Si
´
Ω
|∇u|2
dx
´
Ω
|u|2dx
=
´
Ω
|
Pi
j=1 αj∇uj
|2
dx
´
Ω
|
Pi
j=1 αjuj|2dx
.
Hence
λi(Ω) ≤ lim inf
n→∞
´
Ωn
|
Pi
j=1 αj∇uj
n|2
dx
´
Ωn
|
Pi
j=1 αjuj
n|2dx
≤ lim inf
n→∞
λi(Ωn),
the last inequality being a consequence of the min-max formula on Ωn.
Using the lower semincontinuity and the monotonicity of F, this gives
F(λ1(Ω), . . . , λk(Ω)) ≤ lim inf
n→∞
F(λ1(Ωn), . . . , λk(Ωn)).
If |Ω| c, then adding an open set U to Ω such that U ⊆ D and |Ω ∪ U| = c, we get
a solution for (2.15). This is again a consequence of the monotonicity of F and of the
eigenvalues on inclusions of sets.
Remark 2.2. The monotonicity of the functional F is crucial. From a technical point
of view, this hypothesis is used above in the construction of the optimal set: the limit
set Ω may have a measure strictly lower than c, and so by enlarging it, we get the
solution. When enlarging the set Ω, the eigenvalues do not increase! The functional F
is tailored to behave well during this procedure. Nevertheless, there are situations in
which existence of a solution holds for functionals F that are not required to satisfy
this property. This is the case of functionals depending only on λ1(Ω) and λ2(Ω) (see
[207, Section 6.4]). It is not known whether a general existence result as Theorem 2.1
may hold if the monotonicity assumption on F is dropped and replaced by a weaker
hypothesis.
Remark 2.3. The boundedness of the design region D plays an important role for com-
pactness, in the construction of the optimal set Ω. The only fact we used in the proof
was that H1
0(D) is compactly embedded in L2
(D), so Theorem 2.1 holds with this hy-
pothesis, instead the stricter hypothesis D bounded. Nevertheless, if D is not bounded,
the existence of a solution may fail. For example, in R2
we consider
D =
+∞
[
i=3
B 1
2 − 1
i
(i, 0), c =
π
4
, F(Ω) = λ1(Ω),
where Br(x) is the ball centered at x of radius r.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-33-320.jpg)
![20 | Dorin Bucur
Then the infimum of the functional is λ1(B 1
2
), which is not attained.
For general unbounded design regions D, Theorem 2.1 may not apply. Neverthe-
less, in the particular, and very important, case D = Rd
, the existence result holds.
This issue is discussed in the next section.
Remark 2.4. Depending on F, the optimal set Ω may satisfy some regularity proper-
ties. We refer to Theorem 2.10 in the next section and to Chapter 3.
Remark 2.5. Given an arbitrary sequence of quasi-open subsets of D, the full behavior
of the spectrum for at least one subsequence is completely understood. In fact, it was
proved (see Dal Maso and Mosco [314]), that there exists a subsequence (still denoted
using the same index) and a positive Borel measure µ, absolutely continuous with
respect to the capacity, such that the sequence of resolvent operators RΩn
: L2
(D) →
L2
(D) converges in the operator norm on L2
(D) to Rµ, defined by Rµ(f) = uµ,f
(
−∆uµ,f + µuµ,f = f in D
uµ,f ∈ H1
0(D) ∩ L2
(D, µ)
in the sense
∀ϕ ∈ H1
0(D) ∩ L2
(D, µ)
ˆ
D
∇uµ,f ∇ϕdx +
ˆ
D
uµ,f ϕdµ =
ˆ
D
fϕdx. (2.16)
The function uµ,f is also the unique minimizer in H1
0(D) ∩ L2
(D, µ) of the functional
u 7→
1
2
ˆ
D
|∇u|2
dx +
1
2
ˆ
D
u2
dµ −
ˆ
D
fudx.
Then, λk(µ) are the inverses of the eigenvalues of the positive, self-adjoint and
compact operator Rµ and are defined via the min-max formula
λk(µ) = min
S∈Sk
max
u∈S{0}
´
D
|∇u|2
dx +
´
D
u2
dµ
´
D
u2dx
, (2.17)
where Sk stands for the family of all subspaces of dimension k in H1
0(D) ∩ L2
(D, µ).
As a consequence of the convergence of the resolvent operators, for every k ∈ N,
λk(Ωn) → λk(µ). Of course, from the point of view of the existence theorem 2.1, the
measure µ does not provide a solution. The original proof of Buttazzo-Dal Maso (done
for a larger class of functionals) consisted in replacing the measure µ, obtained from a
minimizing sequence, by a quasi-open set which is, roughly speaking, the union of all
sets of µ-finite measure. This set was proved to be optimal, thanks to the monotonicity
of F.
Remark 2.6. There is a second proof of Theorem 2.1, given in [210], which is based
on the so called weak gamma convergence. It is said that Ωn weakly gamma converges
to Ω if (wΩn
)n converges weakly in H1
0(D) to some function w, and Ω = {w 0}. This](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-34-320.jpg)
![2 Existence results | 21
convergence is compact in the family of quasi-open subsets of D, and the Lebesgue
measure is lower semicontinuous. The key consequence of this convergence is the fol-
lowing property.
– Assume Ωn weakly gamma converges to Ω. For all sequences (unk )k, such that unk ∈
H1
0(Ωnk ) and unk converges weakly in H1
0(D) to some function u, then u has to belong
to H1
0(Ω).
Following the proof of Theorem 2.1, this property implies immediately that
∀k ∈ N, λk(Ω) ≤ lim inf
n→∞
λk(Ωn).
As a consequence, existence in Theorem 2.1 comes by a compactness-semicontinuity
argument.
Remark 2.7. More importantly, the argument of the previous remark works for ev-
ery k ∈ N, since the set Ω is built from the torsion functions and not from the first
eigenfunctions. As a consequence, the existence result can be extended to function-
als depending on the full spectrum. So, if F : RN
→ R is lower semicontinuous in a
suitable sense and non decreasing in each variable, then the existence result proved
in Theorem 2.1 could apply. An example would be
Ω 7→
∞
X
k=1
e−λk(Ω)
−1
.
2.4 Global existence results: the design region is Rd
In this section we deal with the problem
min
n
F λ1(Ω), .., λk(Ω)
: Ω quasi-open, Ω ⊆ Rd
, |Ω| = c
o
(2.18)
The passage from a bounded design region D to Rd
is not trivial. In fact, the com-
pact embedding of H1
0(D) in L2
(D), which played a crucial role in the proof, fails to be
true in Rd
. For example, in the proof of Theorem 2.1, if D = Rd
, the limit functions ui
may all be equal to zero. This occurs, for instance, if the sets Ωn have distances to the
origin tending towards +∞.
From this perspective, a first attempt to solve the existence problem in Rd
was
done in [219] and was based on the concentration compactness principle of Pierre-
Louis Lions. The following result holds (see [204, 207]).
Theorem 2.8. (Bucur) Let (Ωn)n be a sequence of quasi-open sets of Rd
of measure
equal to c. One of the following situations holds.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-35-320.jpg)
![22 | Dorin Bucur
Compactness: There exists a subsequence (Ωnk )k∈N, a sequence of vectors (yk)k∈N ⊆
Rd
and a positive Borel measure µ, vanishing on sets of zero capacity, such that
kRyk+Ωnk
− RµkL(L2(Rd)) → 0 as k → +∞
and
S
k∈N H1
0(Ωnk ) is collectively compactly embedded in L2
(Rd
).
Dichotomy: There exists a subsequence (Ωnk )k∈N and a sequence of subsets Ω̃k ⊆ Ωnk ,
such that
kRΩnk
− RΩ̃k
kL(L2(Rd)) → 0, and Ω̃k = Ω1
k ∪ Ω2
k
with d(Ω1
k , Ω2
k) → ∞ and lim inf
n→∞
|Ωi
k| 0 for i = 1, 2.
In [219], this result was used to prove the existence of a minimizer for λ3. Following
Theorem 2.8, a minimizing sequence (Ωn)n can be either in the compactness case,
in which we get existence, or in the dichotomy case, in which case, the minimizing
sequence can be chosen such that it consists of disconnected sets. In this situation,
the problem is reduced to finding the minimizers of λ1 and λ2 which are known.
In order to use this argument to prove the existence of a minimizer for λ4, it would
be enough to prove that the minimizer for λ3 is, for instance, bounded. In this case,
the dichotomy would lead to a combination of a minimizer of λ3 and a ball. If the min-
imizer of λ3 (that we know its existence) was not bounded, then the union of the min-
imizer and a ball may always have a non-trivial intersection. This would be the case
if the minimizer of λ3 was a dense set in Rd
. Of course, this situation is not expected,
but to exclude it one has to understand some qualitative properties of the minimizers.
The global existence result in Rd
, was proved independently in [206] and [700], by
completely different methods. In [206], the notion of shape subsolution for the torsion
energy (see the next section) was introduced, and it was proved that every such sub-
solution has to be a bounded set with finite perimeter. A second argument showed
that minimizers for (2.18) are shape subsolutions, so they are bounded, hence the
concentration-compactness theorem 2.8 can be used.
The proof in [700] used a surgery result which states that from every set with a
diameter large enough, some parts can be cut out such that after small modifications
and rescaling, the new set has a diameter below some treshold and not larger (low)
eigenvalues. In this way, replacing the minimizing sequence, the existence problem
in Rd
was reduced to the local case of Buttazzo and Dal Maso.
In the next section, we shall give the main ideas of the proof of the global existence
result, using a combination of the two methods: shape subsolutions and surgery. The
following result was proved in [221], as an extension of the surgery result of [700] and
using the subsolution method of [206].
Lemma 2.9. (surgery) For every K, c 0, there exists D, C 0 depending only on K, c
and the dimension d such that for every quasi-open set Ω ⊂ Rd
with |Ω| = c there exists
a quasi-open set Ω̃ with |Ω̃| = c, diam (Ω̃) ≤ D, Per(Ω̃) ≤ C and, if for some k ∈ N it](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-36-320.jpg)
![2 Existence results | 23
holds λk(Ω) ≤ K, then
∀1 ≤ i ≤ k, λi(Ω̃) ≤ λi(Ω). (2.19)
Moreover, if Per(Ω) C the inequalities (2.19) are strict.
We shall give the lines of the proof of this lemma in the next section. Here is the main
consequence.
Theorem 2.10. (Bucur, Mazzoleni, Pratelli) Let F : Rk
→ R be non-decreasing in
each variable and lower semicontinuous. Then problem (2.18) has at least one solution.
If F is strictly increasing in at least one variable, then every solution of (2.18) is a bounded
set with finite perimeter.
Proof. Let (Ωn)n be a minimizing sequence for (2.18). One can choose Ωn such that
λk(Ωn) ≤ K for all k and some suitable value K. Otherwise, the minimum of F would be
formally achieved at (+∞, . . . , +∞), which from the monotonicity assumption implies
that F has to be constant. As a consequence, we can use the surgery Lemma 2.10 and
find a new sequence (Ω̃n) which has a uniformly bounded diameter and satisfies the
measure constraint. Since ∀1 ≤ i ≤ k, λi(Ω̃) ≤ λi(Ω), the monotonicity of F implies that
this new sequence is also minimizing. At this point, we can use the Buttazzo-Dal Maso
Theorem 2.1, up to possible translations of all Ω̃ in the ball centered at the origin of
radius D, and get the existence of an optimal set.
Assume now that Ω is optimal and F is strictly increasing in at least one variable.
If Per(Ω) C, then we have that Ω̃ is a new minimizer with the first k eigenvalues
strictly smaller than those in Ω. Since F is strictly increasing, we are in contradiction
with the optimality of Ω.
2.5 Subsolutions for the torsion energy
In order to explain the main lines of the proof of Lemma 2.9, we recall the notion of
shape subsolutions introduced in [206], which allows us to replace the study of a gen-
eral spectral functional with the study of the torsion energy. Roughly speaking, if a
shape is optimal for a general spectral functional, it may satisfy some sub-optimality
conditions for the torsion energy. From this last condition, one could deduce inter-
esting qualitative properties on the optimal shape, like information on the perimeter
and outer density, which is related to the boundedness. The key argument is that the
variation of an eigenvalue for an arbitrary geometric perturbation of the domain can
be controlled by the variation of the torsion energy, for the same perturbation (see
inequality (2.8)).](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-37-320.jpg)
![24 | Dorin Bucur
Definition 2.11. We say that a quasi-open set Ω ⊂ Rd
is a local shape subsolution for
the torsion energy if there exists η, δ 0 such that for all quasi-open sets A ⊆ Ω with
the property that
´
(wΩ − wA)dx δ we have
E(Ω) + η|Ω| ≤ E(A) + η|A|.
The main result proved by Bucur in [206] is the following.
Theorem 2.12. Assume Ω is a local shape subsolution for the torsion energy. Then Ω is
bounded and has finite perimeter.
In the result above, the diameter is understood as the maximal length of the orthog-
onal projection of Ω on lines. Both the diameter and the perimeter depend only on
|Ω|, η, δ, d.
Proof. (of Theorem 2.12) The proof of the boundedness is a consequence of the follow-
ing Lemma, inspired from the seminal paper of Alt and Caffarelli [25].
Lemma 2.13. Assume Ω is a local shape subsolution for the torsion energy. There exists
r0 0, C0 0 such that for every x0 ∈ Rd
and r ∈ (0, r0)
if sup
x∈Br(x0)
wΩ(x) ≤ C0r then wΩ = 0 on B 1
2 r(x0). (2.20)
The proof of this lemma is classical. We refer the reader to [25], or to [206], for the
specific situation of the torsion function.
In order to gather information on the boundedness of a shape subsolution, one
observes that for every θ 0, there exists δ0 0 depending only on N, θ such that if
wΩ(x0) ≥ θ for some x0 ∈ Rd
, then
ˆ
Bδ(x0)
wΩdx ≥
wΩ(x0)ωd
2
δd
, ∀δ ∈ (0, δ0).
Indeed, for every x0 ∈ Rd
the function x 7→ wΩ(x) + |x−x0|2
2d is subharmonic in Rd
.
Consequently, for every δ 0
θ ≤ wΩ(x0) ≤
1
|Bδ|
ˆ
Bδ(x0)
(w(x) +
|x − x0|2
2d
) dx =
1
|Bδ|
ˆ
Bδ(x0)
wdx +
δ2
2(d + 2)
.
For δ0 sufficiently small, we have ∀ 0 δ ≤ δ0
ˆ
Bδ(x0)
wΩdx ≥
wΩ(x0)ωd
2
δd
.
As a consequence, if Ω is unbounded (or has a large diameter, in the sense that the
Hausdorff measure of the projection of the set Ω on one line is large) we get that the
measure is larger than any constant (depending on the length of the diameter).](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-38-320.jpg)
![26 | Dorin Bucur
Since Ωη is a subsolution for the torsion energy, the diameter and perimeter of Ω̃ are
controlled only by η, c and d.
It is easy to notice that if η is small enough (the precise value will be fixed at the
end), then the first k eigenvalues of Ω̃ are not larger than the corresponding eigenval-
ues on Ω. This is essentially a consequence of inequality (2.8)
λk(Ωη) − λk(Ω) ≤ 4k2
e1/4π
λk(Ωη)λk(Ω)(d+2)/2
[E(Ω) − E(Ωη)]. (2.21)
Indeed, if η is small enough, one can prove (see (2.10)-(2.11)) that λk(Ωη) ≤ 2λk(Ω) and
get
λk(Ωη) − λk(Ω) ≤ Cη,λk(Ω),c,d(|Ω|
2
d − |Ωη|
2
d ). (2.22)
The constant Cη,λk(Ω),c,d is smaller when η and λk(Ω) are smaller. The dependence of
Cη,λk(Ω),c,d on all parameters, including η is explicit. As a consequence, we get
λk(Ωη) + Cη,λk(Ω),c,d|Ωη|
2
d ≤ λk(Ω) + Cη,λk(Ω),c,d|Ω|
2
d , (2.23)
which leads for any value Cη,λk(Ω),c,d ≤ λk(Ω)
c
2
d
to the inequality
λk(Ωη)|Ωη|
2
d ≤ λk(Ω)|Ω|
2
d . (2.24)
This implies that λk(Ω̃) ≤ λk(Ω).
Clearly, inequalities λi(Ω̃) ≤ λi(Ω) also hold for i = 1, . . . , k − 1.
Moreover, the set Ωη being a subsolution for the torsion energy, it is bounded and
has finite perimeter, controlled only by Ω, η and d. This holds as well for the set Ω̃,
with rescaling factors coming from the ratio |Ω|
|Ωη|
.
Provided that the constant η is chosen small enough such that this ratio is not
larger than 2 and that Cη,λk(Ω),c,d ≤ λk(Ω)
c
2
d
, we conclude the proof.
Further remarks
The perimeter constraint. A natural question is to ask if Theorem 2.10 could hold
under a further constraint Per(Ω) ≤ c2. Of course, this question becomes interesting,
as soon as the constant c2 is smaller than the perimeter of the optimal set in Theorem
2.10. In order to deal with these kind of questions, in [221] a second surgery result is
proved, with the purpose of having a finer control of the perimeter. This result asserts,
roughly speaking, that one can also decrease the perimeter of a set if its diameter is
large in Lemma 2.9. For this purpose, the surgery procedure is performed in a different
way.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-40-320.jpg)
![2 Existence results | 27
Lemma 2.14. (surgery of the perimeter) For every K, P, c 0, there exist D 0
depending only on K, P, c and the dimension d, such that for every quasi-open set Ω ⊂
Rd
with |Ω| = c, Per(Ω) ≤ P, there exists a quasi-open set Ω̃ of the same measure, with
diam (Ω̃) ≤ D, Per(Ω̃) ≤ Per(Ω) such that if for some k ∈ N it holds λk(Ω) ≤ K, then
λi(Ω̃) ≤ λi(Ω) for all 1 ≤ i ≤ k.
A consequence of this lemma concerns the following spectral optimization problem
min{F(λ1(Ω), . . . , λk(Ω)) : Ω ⊆ Rd
, |Ω| = c1, Per(Ω) ≤ c2}. (2.25)
Theorem 2.15. Provided that F : Rk
→ R is non-decreasing in each variable and lower
semicontinuous, for every c1, c2 0 such that c2 ≥ Hd−1
(∂B1)
|B1|
d−1
d
c
d−1
d
1 , problem (2.25) has a
solution in the class of measurable sets.
We refer to [330] for details on shape optimization problems on measurable sets.
Roughly speaking, this means that the minimum is attained on a quasi-open set Ω,
for which there exists a measurable set A such that Ω ⊆ A, |A| = c1 and Per(A) ≤ c2
(see [221] for details).
Optimization in specific classes of sets. An interesting task is to search for the ex-
tremal sets of spectral functionals in some specific classes of sets, e.g. the class of
convex subsets of Rd
(satisfying, or not, a constraint on measure, perimeter or diam-
eter), the class of simply connected sets open sets of R2
, the class of N-gones of R2
,
etc. As a general fact, one can notice that the existence question has a much more
direct answer, as soon as those geometric or topological constraints are imposed. For
instance, Theorems 2.1, 2.10 can be rephrased in the class of open convex sets (in any
dimension of the space) or in the class of open sets in R2
whose complement have
at most l connected components (l is a fixed natural number) (see [207, Sections 4.6,
4.7 and Chapters 5, 6]). In the family of convex sets, very interesting phenomena may
occur leading to optimal sets which are locally of polygonal type. We refer the reader
to [622, 623] and Section 3.5 of Chapter 3.
Other boundary conditions, higher order operators. In this chapter we discussed
the existence questions only for functionals depending on the spectrum of the Laplace
operator with Dirichlet boundary conditions. A good question is whether or not similar
results hold for the Laplace operator with other boundary conditions.
Working with different boundary conditions requires us to completely change the
functional framework. For instance, when optimizing spectral functionals associated
with the Neumann Laplacian, similar techniques as in the proof of Theorem 2.1 can
hardly be used. In fact, the functional space one has to use for the Neumann Lapla-
cian is H1
(Ω). For different sets Ω, the spaces H1
(Ω) are not naturally embedded in
a good functional space, unless (uniform) geometric requirements are satisfied by](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-41-320.jpg)
![28 | Dorin Bucur
the different sets. The existence question for general functionals depending on the
spectrum of the Neumann Laplacian is completely open (see [207, Chapter 7]).
For Robin boundary conditions (or for the Steklov problem) one could use the
theory of special functions of bounded variations in order to handle existence, at least
in some specific situations. The regularity of the boundaries of the optimal sets, relies
here on the theory of free discontinuity problems. We refer the reader to the discussion
around Theorem 4.24 in Chapter 4 and to [216, 217] for an introduction to the topic.
For the bi-Laplace operator with Dirichlet boundary conditions, a similar result to
Theorem 2.1 holds true in a bounded design region, while for other boundary condi-
tions or D = Rd
, the question is open.
Asymptotic behavior for large k. An interesting question is to understand the be-
havior of a sequence of solutions of problem (2.5) when k goes to +∞. Only partial
answers are known: in two dimensions of the space, if the measure constraint is re-
placed by a perimeter constraint, then any sequence of optimal domains converges to
the disc, as it was recently proved by Bucur and Freitas. The question is to understand
if a similar result continues to be true for the measure constraint, in any dimension
of the space. A key problem is to prove that all the optimal sets for problem (2.5) are
uniformly bounded, independently on k. Even partial results, asserting that subse-
quences of solutions have a geometric limit, would be of interest.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-42-320.jpg)
![30 | Jimmy Lamboley and Michel Pierre
with uniqueness in both cases up to translations (and sets of zero-capacity). Here, D
’large enough’ means that, when k = 1, it can contain a ball of volume a, and when k = 2, it can contain
two disjoint identical balls whose total volume is a. Thus full regularity holds for the optimal
shape in these two cases. The question remains however open for ’large’ D with k ≥ 3
and for any k with ’small’ D. Then, the regularity analysis of the optimal shapes in (3.1)
is very similar to the analysis of the optimal shapes for the Dirichlet energy, namely
min
Gf (Ω), Ω ⊂ D, |Ω| = a , (3.2)
where f ∈ L∞
(D) is given and
Gf (Ω) =
ˆ
Ω
1
2
|∇uΩ|2
− f uΩ
, uΩ ∈ H1
0(Ω), −∆uΩ = f in Ω. (3.3)
(The solution uΩ of this Dirichlet problem is classically defined when Ω is an open set with finite measure.
As explained in Chapter 2, this definition may be extended to the case when Ω is only a quasi-open set
with finite measure.)
Actually, for these two problems (3.1) and (3.2), the analysis of the regularity fol-
lows the same main steps and offers the following main features. They will provide
the content of Sections 3.2 and 3.3.
1. The situation is easier when the state function is nonnegative ! For the Dirich-
let energy case (3.2), for instance in dimension two, full regularity of the boundary
holds for positive data f, inside D (see [187] and Paragraph 3.2.3.1 below). On the
other hand, even in dimension two, it is easily seen that singularities do necessar-
ily occur at each point of the boundary of the optimal set Ω*
in the neighborhood
of which the state function uΩ* (as defined in (3.3)) changes sign. The change of
sign of uΩ* does imply that its gradient has to be discontinuous and, therefore,
that the boundary cannot be regular near these points. For instance, cusps will
then generally occur in dimension two (see e.g. [509]).
For the eigenvalue problem (3.1), state functions are the k-th eigenfunctions on
Ω*
of the Laplace-Dirichlet operator. Thus the situation (and the analysis) will be
quite different if k = 1 where the first eigenfunction is nonnegative and if k ≥ 2
where the eigenfunction changes sign. This partly explains why we devote the
specific Section 3.2 to Problem (3.1) with k = 1. One more specific feature is that
the problem is then equivalent to a minimization problem where the variables are
functions rather than domains and we are led to a free boundary formulation (see
Paragraph 3.2.1) where one has to understand the regularity of the boundary of
[uΩ* 0]. One can essentially obtain as good of regularity results as one did with
the Dirichlet energy case and nonnegative data f, see [189]. Here we strongly rely
on the seminal paper [25] by Alt-Caffarelli about regularity of free boundaries.
On the other hand, the case k ≥ 2 is far from being so well understood and we will
try to describe what current state of the art is (see Section 3.3).](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-44-320.jpg)
![3 Regularity of optimal spectral domains | 31
2. A first step: regularity of the state function. For the Dirichlet energy case, the
analysis starts by studying the regularity of uΩ* as defined in (3.3). It is proved (see
[188]) that uΩ* is locally Lipschitz continuous on D, for any optimal shape Ω*
and
no matter the sign of uΩ* . This Lipschitz continuity is the optimal regularity we
can expect for uΩ* , as it vanishes on D Ω*
, and is expected to have a non vanish-
ing gradient on ∂Ω*
from inside Ω*
. As expected, the proof in the case where uΩ*
changes sign is much more involved and requires for instance the Alt-Caffarelli-
Friedman Monotonicity Lemma (proved in [26], [245], see Lemma 3.36 below).
For the optimal eigenvalue problem (3.1) with k = 1, it can be proved as well that
the corresponding eigenfunction on Ω*
is locally Lipschitz continuous on D (see
Theorem 3.16). For k ≥ 2 and D = Rd
, it has been proved in [222] that one of the k-th
eigenfunctions is Lipschitz continuous (see Theorem 3.35) (note that the optimal
eigenvalue is generally expected to be of multiplicity higher than once). However,
in the case where D is bounded and k ≥ 2, the problem is still not understood. The
main difference is that, when D = Rd
, Problem (3.1) is equivalent to the penalized
version
min
n
λk(Ω) + µ|Ω|, Ω ⊂ Rd
o
, (3.4)
for some convenient µ ∈ (0, ∞) (see Proposition 3.33). More regularity information
may then be derived on optimal state functions for penalized versions (see below).
3. Penalized versions. In order to obtain information on the regularity of Ω*
or uΩ* ,
we consider admissible perturbations of Ω*
and use their minimization proper-
ties. Obviously, there is more freedom in choosing perturbations on the penalized
version (3.4) where the volume constraint |Ω| = a is relaxed, rather than on the
constrained initial version (3.1). The analysis of (3.1) when k = 1 starts by showing
that (3.1) is equivalent to the penalized version
min
λ1(Ω) + µ[|Ω| − a]+
, Ω ⊂ D , (3.5)
for µ large enough (see Proposition 3.7). Analysis of the regularity may then be
more easily made on the optimal shapes of (3.5). In Paragraph 3.2.3.2, we make an
heuristic analysis of this “exact penalty” property for general optimization prob-
lems where not only the penalized version converges to the constrained problem
as the penalization coefficient µ → ∞, but more precisely that the two prob-
lems are equivalent for µ large enough. Optimal such factors µ play the role of La-
grange multipliers. This approach is used again in a local way in Paragraph 3.2.3.3,
to prove that the ’pseudo’-Lagrange multiplier does not vanish (see Proposition
3.24). It is also used in Chapter 7 of this book to study the regularity of optimal
shapes for similar functionals (see Step 5 in the proof of Theorem 7.13).
4. How to obtain the regularity of the boundary of Ω*
? Knowing that the state
function is Lipschitz continuous is a first main step in the study of the regularity
of the boundary of the optimal set, but obviously not sufficient.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-45-320.jpg)
![32 | Jimmy Lamboley and Michel Pierre
For example when k = 1, this boundary can be seen as the boundary of the set
[uΩ* 0]. If we were in a regular situation (say if u were C1
on Ω*), then knowing
that the gradient of uΩ* does not vanish at the boundary would imply regularity
of this boundary by the implicit function theorem.
Indeed, the next main step is (heuristically) to prove that the gradient of the state
function does not degenerate at the boundary. This is what is done and then used
in Paragraph 3.2.3.3 for the optimal sets of (3.1) when k = 1. Full regularity of the
boundary is proved in dimension two and regularity of the reduced boundary is
proved in any dimension (see Theorem 3.20). Here we strongly rely on the seminal
paper [25] by Alt-Caffarelli as explained in details in Section 3.2.3.1. Note that it is
also used in Chapter 7 of this book as mentioned at the end of Point 3 above. Noth-
ing like this is known when k ≥ 2. It is already a substantial piece of information
to sometimes know that Ω*
is an open set ! (see Section 3.3).
In Section 3.4, we partially analyze the regularity of Ω*
solution of (3.1) up to the
boundary of the box D, when k = 1. We notice in particular that it is natural to expect
the contact to be tangential (although this is not proved anywhere as far as we know),
but we cannot expect in general that the contact will be very smooth; we prove that
when D is a strip (too narrow to contain a disc of volume a), the optimal shape is C1,1/2
and not C1,1/2+ε
with ε 0. In order to show that this behavior is not exceptional and
is not only due to the presence of a box constraint, we show that a similar property is
valid for solutions to the problem
min
λ2(Ω), Ω open and convex, |Ω| = a .
This last problem enters the general framework of convexity constraints, which is
quite challenging from the point of view of calculus of variations. We conclude this
chapter with Section 3.5 where we discuss some problems in this framework. They are
of the form
min
J(Ω), Ω open and convex ,
where J involves λ1, and possibly other geometrical quantities (such as the volume |Ω|
or the perimeter P(Ω)), and which lead to singular optimal shapes, such as polygons
(in dimension 2). Thanks to the convexity constraint, we are allowed to consider the
question of maximizing the perimeter and/or the first Dirichlet eigenvalue, and in this
direction we discuss a few recent results about the reverse Faber-Krahn inequality.
Remark 3.1. The question of regularity could also be considered for the following
optimization problems:
min
λk(Ω), Ω ⊂ D, P(Ω) = p , min
P(Ω) + λk(Ω), Ω ⊂ D, |Ω| = a
where P denotes the perimeter (in the sense of geometric measure theory), and D is
either a bounded smooth box, or Rd
. In these cases, it has been shown in [329, 330]](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-46-320.jpg)
![34 | Jimmy Lamboley and Michel Pierre
section on the minimization of λ1(Ω). If Ω is a connected open set, then uΩ 0 on Ω.
This is a consequence of the maximum principle applied to −∆uΩ = λ1(Ω)uΩ ≥ 0 on Ω.
This extends (quasi-everywhere) to the case when Ω is a quasi-connected quasi-open
set, but the proof requires a little more computation.
Since Ω 7→ λ1(Ω) is nonincreasing with respect to inclusion, any solution of (3.6) is
also solution of
min
λ1(Ω), Ω ⊂ D, Ω quasi − open, |Ω| ≤ a . (3.9)
The converse is true in most situations, in particular if D is connected, see Remark 3.6,
Corollary 3.18 and the discussion in Section 3.2.4.1. Note that it may happen that if D
is not connected, then a solution to (3.9) does not satisfy |Ω| = a.
We will first consider Problem (3.9) and this will nevertheless provide a complete
understanding of (3.6). We start by proving that (3.9) is equivalent to a free boundary
problem.
Proposition 3.4. 1. Let Ω*
be a quasi-open solution of the minimization problem
(3.9) and let u = uΩ* . Then
ˆ
D
|∇u|2
= min
ˆ
D
|∇v|2
; v ∈ H1
0(D);
ˆ
D
v2
= 1, |Ωv| ≤ a
. (3.10)
2. Let u be solution of the minimization problem (3.10). Then Ωu is solution of (3.9).
Proof. For the first point, we choose v ∈ H1
0(D) with |Ωv| ≤ a and we apply (3.9) to
Ω = Ωv. This gives
´
D
|∇u|2
= λ1(Ω*
) ≤ λ1(Ωv) and we use the property (3.7) for λ1(Ωv)
so that ˆ
D
|∇u|2
≤ min
ˆ
D
|∇v|2
; v ∈ H1
0(D),
ˆ
D
v2
= 1, |Ωv| ≤ a
.
Equality holds since u ∈ H1
0(Ω*
) ⊂ H1
0(D), and |Ωu| = |Ω*
| ≤ a.
For the second point, let u be a solution of (3.10). Then, |Ωu| ≤ a,
´
D
u2
= 1. Let
Ω ⊂ D quasi-open with |Ω| ≤ a and let uΩ as in Definition 3.2. Then
λ1(Ωu) ≤
ˆ
D
|∇u|2
≤
ˆ
D
|∇uΩ|2
= λ1(Ω).
Remark 3.5. We will now work with the functional problem (3.10) rather than (3.9).
Note that if D is bounded (or with finite measure), then existence of the minimum u
follows easily from the compactness of H1
0(D) into L2
(D) applied to a minimizing se-
quence (that we may assume to be weakly convergent in H1
0(D) and strongly in L2
(D)).
Remark 3.6. Two different situations may occur. If D is connected and Ω*
solves (3.9),
then a*
:= |[uΩ* 0]| = a and Ω*
= [uΩ* 0]. If D is not connected, it may happen](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-48-320.jpg)
![3 Regularity of optimal spectral domains | 35
that a*
a and therefore uΩ* 0 on some of the connected components of D and
identically zero on the others.
Indeed, if a*
a, then for all balls B ⊂ D with measure less than a − a*
and all
φ ∈ H1
0(B), we may choose v = v(t) = (u + tφ)/ku + tφkL2(D) with u := uΩ* ≥ 0 in (3.10).
Writing that the derivative at t = 0 of t 7→
´
D
|∇v(t)|2
vanishes gives
ˆ
D
∇u∇φ = λa
ˆ
D
u φ with λa :=
ˆ
D
|∇u|2
,
and this implies : −∆u = λau in D. The strict maximum principle implies that, in each
connected component of D, either u 0 or u ≡ 0. If D is connected, we get a contra-
diction since a |D|. Therefore necessarily a*
= a if D is connected.
We refer to Corollary 3.18 and Proposition 3.29 for a complete description of the
regularity when D is not connected.
3.2.2 Existence and Lipschitz regularity of the state function
3.2.2.1 Equivalence with a penalized version
We will first prove that (3.10) is equivalent to a penalized version.
Proposition 3.7. Assume |D| +∞. Let u be a solution of (3.10) and λa :=
´
D
|∇u|2
.
Then, there exists µ 0 such that
ˆ
D
|∇u|2
≤
ˆ
D
|∇v|2
+ λa
1 −
ˆ
D
v2
+
+ µ [|Ωv| − a]
+
, ∀v ∈ H1
0(D). (3.11)
Remark 3.8. Given a quasi-open set Ω ⊂ D, and choosing v = uΩ in (3.11), we obtain
the penalized ’domain’ version of (3.9), where Ω*
is solution of (3.9)
λ1(Ω*
) ≤ λ1(Ω) + µ[|Ω] − a]+
, ∀ Ω ⊂ D, Ω quasi − open. (3.12)
Proof of Proposition 3.7.. Note first that, by definition of u and of λa, for all v ∈ H1
0(D)
with |Ωv| ≤ a, we have
´
D
|∇v|2
− λa
´
D
v2
≥ 0, or
ˆ
D
|∇u|2
≤
ˆ
D
|∇v|2
+ λa
1 −
ˆ
D
v2
. (3.13)
Let us now denote by Jµ(v) the right-hand side of (3.11) and let uµ be a minimizer of
Jµ(v) for v ∈ H1
0(D) (its existence follows by compactness of H1
0(D) into L2
(D), see also
Remark 3.5). Up to replacing uµ by |uµ|, we may assume uµ ≥ 0. Using that Jµ(uµ) ≤
Jµ(uµ/kuµk2), we also deduce that kuµk2
2 =
´
D
u2
µ ≤ 1.
For the conclusion of the proposition, it is sufficient to prove |Ωuµ | ≤ a since then
Jµ(uµ) ≤ Jµ(u) =
ˆ
D
|∇u|2
≤ Jµ(uµ),](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-49-320.jpg)
![36 | Jimmy Lamboley and Michel Pierre
where this last inequality comes from (3.13).
In order to obtain a contradiction, assume that |Ωuµ | a and introduce ut
:=
(uµ − t)+
. Then Jµ(uµ) ≤ Jµ(ut
). This implies, using |Ωut | a for t small, that
ˆ
[0uµ t]
|∇uµ|2
+ µ [0 uµ t] ≤ λa
ˆ
[0uµ t]
u2
µ + 2tλa
ˆ
D
uµ.
Using the coarea formula (see e.g. [370], [432]), this may be rewritten for t ≤ t0 ≤
p
µ/λa as
ˆ t
0
ds
ˆ
[uµ=s]
|∇uµ| +
µ − λas2
|∇uµ|
dHd−1
≤ 2tλa
ˆ
D
uµ ≤ 2tλa|Ωuµ |1/2
.
But the function x ∈ (0, ∞) 7→ x + (µ − λas2
)x−1
∈ [0, ∞) is bounded from below
by 2
p
µ − λas2 and also by 2
q
µ − λat2
0 as soon as s2
≤ t2
≤ t2
0 ≤ µ/λa. Therefore, it
follows that
∀ t ∈ [0, t0), 2
q
µ − λat2
0
ˆ t
0
ˆ
[uµ=s]
dHd−1
≤ 2tλa|Ωuµ |1/2
. (3.14)
We now use the isoperimetric inequality:
´
[uµ=s]
dHd−1
≥ C(d) [uµ s]
d−1
d
. We divide
the inequality by t and we let t → 0, then t0 → 0, to deduce
2
√
µ C(d)|Ωuµ |
d−1
d ≤ 2λa|Ωµ|1/2
, and finally 2
√
µ C(d) a
d−2
2d ≤ 2λa.
Thus, if d ≥ 2, |Ωuµ | a is impossible if µ µ*
:= λ2
aC(d)−2
a(2−d)/d
. Therefore the
conclusion of Proposition 3.7 holds for any µ µ*
.
If d = 1, we have
√
µC(1) ≤ λa|Ωµ|1/2
. On the other hand, by definition of uµ we
also have |Ωuµ | ≤ a + λ1(Ω1)/µ for some fixed Ω1 ⊂ D with |Ω1| = a. We deduce an
upper bound for µ as well.
Remark 3.9. With respect to the heuristic remarks made in Paragraph 3.2.3.2, it is
interesting to notice that our problem here is not in a ’differentiable setting’. However,
we do perform some kind of differentiation in the direction of the perturbations t 7→
(uµ − t)+
. This provides the upper bound µ*
on µ which plays the role of a Lagrange
multiplier. This remark is a little more detailed in Paragraph 3.2.3.2. Note that µ*
does
not depend on |D|. The assumption |D| ∞ was used only to prove existence of the
minimizer uµ.
Remark 3.10 (Sub- and super-solutions). Note that to prove Proposition 3.7, we only
use perturbations of the optimal domain Ωu from inside. This means that the same
result is valid for shape subsolutions where (3.10) is assumed only for functions v for
which Ωv ⊂ Ωu.
Next, we will prove Lipschitz continuity of the functions u solutions of the penal-
ized problem (3.11). Interestingly, Lipschitz continuity will hold for super-solutions of
(3.11) which are defined when the inequality (3.11) is valid only for perturbations from
outside, i.e. such that Ωu ⊂ Ωv.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-50-320.jpg)
![3 Regularity of optimal spectral domains | 37
3.2.2.2 A general sufficient condition for Lipschitz regularity
We now state a general result to prove Lipschitz regularity of functions independently
of shape optimization. It applies to signed functions as well and will be used again in
the minimization of the k-th eigenvalue.
Proposition 3.11. Let U ∈ H1
0(D), bounded and continuous on D and let ω := {x ∈
D; U(x)=
6 0}. Assume ∆U is a measure such that ∆U = g on ω with g ∈ L∞
(ω) and
|∆|U|| (B(x0, r)) ≤ Crd−1
(3.15)
for all x0 ∈ D with B(x0, 2r) ⊂ D, r ≤ 1 and U(x0) = 0. Then U is locally Lipschitz
continuous on D. If moreover D = Rd
, then U is globally Lipschitz continuous.
Remark 3.12. Note that if U is locally Lipschitz continuous on D with ∆U ≥ 0, then
for a test function φ with
φ ∈ C∞
0 (B(x0, 2r)), B(x0, 2r) ⊂ D, 0 ≤ φ ≤ 1,
φ ≡ 1 on B(x0, r), k∇φkL∞(B) ≤ C/r,
(3.16)
we have
∆U(B(x0, r)) ≤
ˆ
D
φd(∆U) = −
ˆ
D
∇φ∇U ≤
k∇UkL∞ |Ωφ|k∇φkL∞ ≤ Ck∇UkL∞ rd−1
.
This indicates that the estimate (3.15) is essentially a necessary condition for the Lips-
chitz continuity of U. This theorem states that the converse holds in some cases which
are relevant for our analysis as it will appear in the next paragraph.
Remark 3.13. In the proof of Proposition 3.11, as in [188], we will use the following
identity which is useful to estimate the variation of functions:
∂B(x0,r)
U(x)dσ(x) − U(x0) = C(d)
ˆ r
0
s1−d
ˆ
B(x0,s)
d(∆U)
ds. (3.17)
This is easily proved for regular functions U by integration in s of
d
ds ∂B(0,1)
U(x0 + sξ)dσ(ξ) =
∂B(0,1)
∇U(x0 + sξ) · ξ = C(d)s1−d
ˆ
B(x0,s)
∆U,
which implies that for a.e. 0 r1 r2,
∂B(x0,r2)
U(x)dσ(x) −
∂B(x0,r1)
U(x)dσ(x) = C(d)
ˆ r2
r1
s1−d
ˆ
B(x0,s)
∆U
ds.
It extends to functions U ∈ H1
(D) where ∆U is a measure with
´ r
0
s1−d
´
B(x0,s)
d(|∆U|)ds ∞. We may then consider that U is precisely defined at x0
as:
U(x0) = lim
r→0+
∂B(x0,r)
U(x)dσ(x), (3.18)](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-51-320.jpg)
![3 Regularity of optimal spectral domains | 39
We deduce, using (3.20) again and the representation (3.18):
|U(x0) − U(y0)| ≤
ffl
∂B(x0,r)
U −
ffl
∂B(y0,r)
U + C r ≤
ffl
∂B(0,r)
|U(x0 + ξ) − U(y0 + ξ)|dσ(ξ) + C r.
But by continuity of the trace operator from H1
(B(0, r)) into L1
(B(0, r)), this implies
|U(x0) − U(y0)| ≤ C(r)kU(x0 + .) − U(y0 + .)kH1(B(0,r)) + C r.
Thus
lim sup
y0→x0
|U(x0) − U(y0)| ≤ C r.
Since this is valid for all r sufficiently small, continuity of U at x0 follows and therefore
continuity on D as well.
Remark 3.15. Looking at the proof, we easily see that the assumptions could be weak-
ened in Lemma 3.14: U ∈ W1,1
0 (D) would be sufficient and rd−1
could be replaced in
(3.20) by rd−2
ε(r) with ε(r)/r integrable on (0, 1).
3.2.2.3 Lipschitz continuity of the optimal eigenfunction
Theorem 3.16. Let u be a solution of (3.10). Then u is locally Lipschitz continuous on
D.
Proof. Up to replacing u by |u|, we may assume that u ≥ 0. We will first show that
U = u satisfies the assumptions of Lemma 3.14. It will follow that u is continuous on
D. Therefore, we will have −∆u = λau on the open set ω = [u 0] (see Remark 3.3).
Then we will prove (see also Remark 3.17 below) that
− ∆u ≤ λau in D. (3.21)
This will imply that ∆u is a measure and also, by an easy bootstrap that u ∈ L∞
(D).
Thus the assumptions of Proposition 3.11 will be satisfied and local Lipschitz continu-
ity on D will follow.
By Proposition 3.7, u is also solution of Problem (3.11). We apply this inequality
with v = u + tφ, t 0, φ ∈ H1
0(D). Then
0 ≤
ˆ
D
2∇u∇φ + t|∇φ|2
+ λa
h
−2uφ − tφ2
i+
+
µ
t
|Ωu+tφ| − a
+
. (3.22)
Choosing first φ = −pn(u)ψ where ψ ∈ C∞
0 (D), ψ ≥ 0, and pn(r) = min{r+
/n, 1},
we obtain with qn(r) =
´ r
0
pn(s)ds and after letting t → 0 (note that |Ωu+tφ| = |Ωu| ≤ a)
0 ≤
ˆ
D
−2p′n(u)|∇u|2
ψ − 2∇qn(u)∇ψ + 2λaupn(u)ψ.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-53-320.jpg)
![40 | Jimmy Lamboley and Michel Pierre
Note that upn(u) → u+
= u, qn(u) → u+
= u in a nondecreasing way as n increases
to +∞. Using p′n(u)|∇u|2
≥ 0, we obtain at the limit that ∆u + λau ≥ 0 in the sense of
distributions in D, whence (3.21).
Choosing φ ∈ C∞
0 (D)+
in (3.22) leads to −2
´
D
∇u∇φ ≤
´
D
t|∇φ|2
+ µ
t |Ωφ| or
2h∆u + λau, φi ≤
ˆ
D
2λauφ + t|∇φ|2
+
µ
t
|Ωφ|. (3.23)
Minimizing over t ∈ (0, ∞) gives
h∆u + λa, φi ≤
ˆ
D
λauφ + k∇φkL2 [µ|Ωφ|]
1/2
. (3.24)
Let now x0 ∈ D such that B(x0, 2r) ⊂ D and let φ ∈ C∞
0 (B(x0, 2r))+
as in (3.16). Using
also u ∈ L∞
, we deduce that
|∆u| B(x0, r)
≤ (∆u + λau) (B(x0, r)
+ λa
ˆ
B(x0,r)
u ≤ Crd−1
,
whence the estimate (3.15).
Remark 3.17. Here, we use the positivity of u. Actually, u is an eigenfunction for the
eigenvalue λa on Ωu. Since Ωu is open, we know that ∆u + λau = 0 on Ωu (see Remark
3.3). Since u ≥ 0, one can prove that ∆u + λau ≥ 0 on D. To prove this, use the test
functions φ = −pn(u)ψ which satisfy Ωφ ⊂ Ωu and therefore belong to H1
0(Ωu). Thus
applying (3.8) in Remark 3.3 with this φ is sufficient (and we finish as above).
This positivity of the measure ∆u + λau allows to directly estimate the mass of
|∆u| on balls only with the information (3.24). This will not be the case when dealing
with k-th eigenfunctions when k ≥ 2 (see the remarks and comments on the use of the
Monotonicity Lemma 3.36).
Let us now state a corollary of Proposition 3.16 for the initial actual shape optimization
problem (3.6).
Corollary 3.18. Assume D is open and with finite measure. Then there exists an open
set Ω*
which is solution of (3.6). Moreover, for any (quasi-open) solution Ω*
of (3.6), uΩ*
is locally Lipschitz continuous on D. If D is connected, then all solutions Ω*
of (3.6) are
open.
Remark 3.19. If D is not connected, then it may happen that Ω*
is not open: we refer
for instance to Example 3.28. However uΩ* is always locally Lipschitz continuous. Let
us mention that the existence of an optimal open set for (3.6) had first been proved
in [469]. A different penalization was used and it was proved that the corresponding
state function converged to a Lipschitz optimal eigenfunction.](https://image.slidesharecdn.com/23743206-250519121526-64ecb357/85/Shape-Optimization-And-Spectral-Theory-1st-Edition-Antoine-Henrot-54-320.jpg)
Shape Optimization And Spectral Theory 1st Edition Antoine Henrot
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- 5. Antoine Henrot (Ed.) Shape optimization and spectral theory Contributors Pedro R.S. Antunes Mark Ashbaugh Virginie Bonnaillie-Noël Lorenzo Brasco Dorin Bucur Giuseppe Buttazzo Guido De Philippis Pedro Freitas Alexandre Girouard Bernard Helffer James Kennedy Jimmy Lamboley Richard S. Laugesen Edouard Oudet Michel Pierre Iosif Polterovich Bartłomiej A. Siudeja Bozhidar Velichkov
- 7. Antoine Henrot (Ed.) Shape optimization and spectral theory | Managing Editor: Agnieszka Bednarczyk-Drąg Associate Editor: Filippo A. E. Nuccio Mortarino Majno di Capriglio Language Editor: Nick Rogers
- 8. Complimentary Copy Not for Sale ISBN 978-3-11-055085-6 e-ISBN 978-3-11-055088-7 This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 license. For details go to http://creativecommons.org/licenses/by-nc-nd/3.0/. Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. © 2017 Antoine Henrot and Chapters’ Contributors, published by de Gruyter Open Published by De Gruyter Open Ltd, Warsaw/Berlin Part of Walter de Gruyter GmbH, Berlin/Boston The book is published with open access at www.degruyter.com. Cover illustration: © Bartlomiej Siudeja Managing Editor: Agnieszka Bednarczyk-Drąg Associate Editor: Filippo A. E. Nuccio Mortarino Majno di Capriglio Language Editor: Nick Rogers www.degruyteropen.com
- 9. Contents Antoine Henrot 1 Introduction | 1 1.1 General introduction | 1 1.2 Content of the book | 1 1.3 Balls and union of balls | 9 1.4 Notation | 11 Dorin Bucur 2 Existence results | 13 2.1 Setting the problem | 13 2.2 The spectrum on open and quasi-open sets | 15 2.3 Existence results | 18 2.4 Global existence results | 21 2.5 Subsolutions for the torsion energy | 23 Jimmy Lamboley and Michel Pierre 3 Regularity of optimal spectral domains | 29 3.1 Introduction | 29 3.2 Minimization for λ1 | 33 3.2.1 Free boundary formulation | 33 3.2.2 Existence and Lipschitz regularity of the state function | 35 3.2.3 Regularity of the boundary | 41 3.2.4 Remarks and perspectives | 50 3.3 Minimization for λk | 53 3.3.1 Penalized is equivalent to constrained in Rd | 54 3.3.2 A Lipschitz regularity result for optimal eigenfunctions | 54 3.3.3 More about k = 2 | 59 3.4 Singularities due to the box or the convexity constraint | 62 3.4.1 Regularity for partially overdetermined problem | 63 3.4.2 Minimization of λ1 in a strip | 65 3.4.3 Minimization of λ2 with convexity constraint | 67 3.5 Polygons as optimal shapes | 69 3.5.1 General result about the minimization of a weakly concave functional | 70 3.5.2 Examples | 71 3.5.3 Remarks on the higher dimensional case | 74
- 10. Dorin Bucur, Pedro Freitas, and James Kennedy 4 The Robin problem | 78 4.1 Introduction | 78 4.2 Basic properties of the Robin Laplacian | 80 4.2.1 Domain monotonicity and rescaling | 83 4.3 A picture of Robin eigencurves | 84 4.3.1 Robin eigencurves in one dimension | 85 4.3.2 Robin eigencurves in higher dimensions | 86 4.4 Asymptotic behaviour of the eigenvalues | 89 4.4.1 Large positive values of the boundary parameter | 89 4.4.2 Large negative values of the boundary parameter | 90 4.5 Isoperimetric inequalities and other eigenvalue estimates | 99 4.5.1 Positive parameter: Faber–Krahn and other inequalities | 99 4.5.2 Negative parameter | 108 4.6 The higher eigenvalues | 113 4.6.1 The second eigenvalue | 114 4.6.2 Higher eigenvalues (positive boundary parameter) | 114 4.6.3 Higher eigenvalues (negative boundary parameter) | 118 Alexandre Girouard and Iosif Polterovich 5 Spectral geometry of the Steklov problem | 120 5.1 Introduction | 120 5.1.1 The Steklov problem | 120 5.1.2 Motivation | 121 5.1.3 Computational examples | 122 5.1.4 Plan of the chapter | 123 5.2 Asymptotics and invariants of the Steklov spectrum | 124 5.2.1 Eigenvalue asymptotics | 124 5.2.2 Spectral invariants | 125 5.3 Spectral asymptotics on polygons | 127 5.3.1 Spectral asymptotics on the square | 127 5.3.2 Numerical experiments | 129 5.4 Geometric inequalities for Steklov eigenvalues | 130 5.4.1 Preliminaries | 130 5.4.2 Isoperimetric upper bounds for Steklov eigenvalues on surfaces | 131 5.4.3 Existence of maximizers and free boundary minimal surfaces | 136 5.4.4 Geometric bounds in higher dimensions | 138 5.4.5 Lower bounds | 139 5.4.6 Surfaces with large Steklov eigenvalues | 140 5.5 Isospectrality and spectral rigidity | 141 5.5.1 Isospectrality and the Steklov problem | 141 5.5.2 Rigidity of the Steklov spectrum: the case of a ball | 142
- 11. 5.6 Nodal geometry and multiplicity bounds | 143 5.6.1 Nodal domain count | 143 5.6.2 Geometry of the nodal sets | 144 5.6.3 Multiplicity bounds for Steklov eigenvalues | 146 Richard S. Laugesen and Bartłomiej A. Siudeja 6 Triangles and Other Special Domains | 149 6.1 Introduction | 149 6.2 Variation, notation, normalization, majorization | 149 6.3 Lower bounds by symmetrization | 152 6.3.1 Dirichlet eigenvalues | 152 6.3.2 Mixed Dirichlet-Neumann eigenvalues | 157 6.4 Lower bounds by unknown trial functions | 161 6.4.1 Illustration of the method | 161 6.4.2 Dirichlet eigenvalues | 163 6.4.3 Neumann eigenvalues | 165 6.5 Lower bounds by other methods | 166 6.5.1 Spectral gap for triangles | 166 6.5.2 High eigenvalues for rectangles | 166 6.6 Sharp Poincaré inequality and rigorous numerics | 167 6.7 Upper bounds: trial functions | 169 6.7.1 Dirichlet eigenvalues | 169 6.7.2 Neumann eigenvalues | 174 6.8 Rectangles | 176 6.9 Equilateral triangles | 177 6.9.1 Dirichlet eigenvalues | 178 6.9.2 Neumann eigenvalues | 178 6.10 Isosceles triangles | 181 6.10.1 Dirichlet eigenvalues | 182 6.10.2 Neumann eigenvalues | 183 6.11 Right triangles | 185 6.11.1 Dirichlet eigenvalues | 185 6.11.2 Neumann eigenvalues | 186 6.11.3 Mixed Dirichlet–Neumann | 187 6.12 Inverse problem — can one hear the shape of a triangular drum? | 187 6.13 Structure of eigenfunctions on special domains | 188 6.13.1 Multiplicity | 188 6.13.2 Hot spots | 189 6.13.3 Number of nodal domains | 191 6.13.4 Boundary sign-changing for eigenfunctions. | 196 6.14 Conjectures for general domains | 198
- 12. Lorenzo Brasco and Guido De Philippis 7 Spectral inequalities in quantitative form | 201 7.1 Introduction | 201 7.1.1 The problem | 201 7.1.2 Plan of the Chapter | 203 7.1.3 An open issue | 204 7.2 Stability for the Faber-Krahn inequality | 204 7.2.1 A quick overview of the Dirichlet spectrum | 204 7.2.2 Semilinear eigenvalues and torsional rigidity | 205 7.2.3 Some pioneering stability results | 207 7.2.4 A variation on a theme of Hansen and Nadirashvili | 212 7.2.5 The Faber-Krahn inequality in sharp quantitative form | 219 7.2.6 Checking the sharpness | 226 7.3 Intermezzo: quantitative estimates for the harmonic radius | 227 7.4 Stability for the Szegő-Weinberger inequality | 232 7.4.1 A quick overview of the Neumann spectrum | 232 7.4.2 A two-dimensional result by Nadirashvili | 233 7.4.3 The Szegő-Weinberger inequality in sharp quantitative form | 237 7.4.4 Checking the sharpness | 241 7.5 Stability for the Brock-Weinstock inequality | 247 7.5.1 A quick overview of the Steklov spectrum | 247 7.5.2 Weighted perimeters | 249 7.5.3 The Brock-Weinstock inequality in sharp quantitative form | 251 7.5.4 Checking the sharpness | 253 7.6 Some further stability results | 254 7.6.1 The second eigenvalue of the Dirichlet Laplacian | 254 7.6.2 The ratio of the first two Dirichlet eigenvalues | 259 7.6.3 Neumann vs. Dirichlet | 269 7.7 Notes and comments | 270 7.7.1 Other references | 270 7.7.2 Nodal domains and Pleijel’s Theorem | 271 7.7.3 Quantitative estimates in space forms | 272 7.8 Appendix | 273 7.8.1 The Kohler-Jobin inequality and the Faber-Krahn hierarchy | 273 7.8.2 An elementary inequality for monotone functions | 275 7.8.3 A weak version of the Hardy-Littlewood inequality | 277 7.8.4 Some estimates for convex sets | 279 Mark S. Ashbaugh 8 Universal Inequalities for the Eigenvalues of the Dirichlet Laplacian | 282 8.1 Introduction | 282 8.2 Proof of the Main Inequality: Yang1 | 287
- 13. 8.3 The Other Main Inequalities and their Proofs: PPW, HP, and Yang2 | 297 8.4 The Hierarchy of Inequalities: PPW, HP, Yang | 299 8.5 Asymptotics and Explicit Inequalities | 305 8.6 Further Work | 315 8.7 History | 320 Giuseppe Buttazzo and Bozhidar Velichkov 9 Spectral optimization problems for Schrödinger operators | 325 9.1 Existence results for capacitary measures | 326 9.2 Existence results for integrable potentials | 333 9.3 Existence results for confining potentials | 347 Virginie Bonnaillie-Noël and Bernard Helffer 10 Nodal and spectral minimal partitions – The state of the art in 2016 – | 353 10.1 Introduction | 353 10.2 Nodal partitions | 354 10.2.1 Minimax characterization | 354 10.2.2 On the local structure of nodal sets | 355 10.2.3 Weyl’s theorem | 357 10.2.4 Courant’s theorem and Courant sharp eigenvalues | 359 10.2.5 Pleijel’s theorem | 360 10.2.6 Notes | 363 10.3 Courant sharp cases: examples | 363 10.3.1 Thin domains | 363 10.3.2 Irrational rectangles | 364 10.3.3 Pleijel’s reduction argument for the rectangle | 365 10.3.4 The square | 366 10.3.5 Flat tori | 367 10.3.6 The disk | 368 10.3.7 Circular sectors | 370 10.3.8 Notes | 370 10.4 Introduction to minimal spectral partitions | 371 10.4.1 Definition | 371 10.4.2 Strong and regular partitions | 371 10.4.3 Bipartite partitions | 372 10.4.4 Main properties of minimal partitions | 373 10.4.5 Minimal spectral partitions and Courant sharp property | 375 10.4.6 On subpartitions of minimal partitions | 376 10.4.7 Notes | 377 10.5 On p-minimal k-partitions | 377
- 14. 10.5.1 Main properties | 377 10.5.2 Comparison between different p’s | 377 10.5.3 Examples | 379 10.5.4 Notes | 380 10.6 Topology of regular partitions | 381 10.6.1 Euler’s formula for regular partitions | 381 10.6.2 Application to regular 3-partitions | 381 10.6.3 Upper bound for the number of singular points | 382 10.6.4 Notes | 383 10.7 Examples of minimal k-partitions | 383 10.7.1 The disk | 383 10.7.2 The square | 383 10.7.3 Flat tori | 385 10.7.4 Circular sectors | 386 10.7.5 Notes | 387 10.8 Aharonov-Bohm approach | 387 10.8.1 Aharonov-Bohm operators | 387 10.8.2 The case when the fluxes are 1/2 | 388 10.8.3 Nodal sets of K-real eigenfunctions | 389 10.8.4 Continuity with respect to the poles | 390 10.8.5 Notes | 393 10.9 On the asymptotic behavior of minimal k-partitions | 394 10.9.1 The hexagonal conjecture | 394 10.9.2 Lower bounds for the length | 395 10.9.3 Magnetic characterization and lower bounds for the number of singular points | 396 10.9.4 Notes | 397 Pedro R. S. Antunes and Edouard Oudet 11 Numerical results for extremal problem for eigenvalues of the Laplacian | 398 11.1 Some tools for global numerical optimization in spectral theory | 399 11.1.1 An historical approach: Genetic algorithm and Voronoi cells | 399 11.1.2 Smooth profiles with few parameters | 400 11.1.3 A fundamental complexity reduction: optimal connected components | 401 11.2 Numerical approach using the MFS | 402 11.3 The menagerie of the spectrum | 406 11.4 Open problems | 408 Bibliography | 413 Index | 462
- 15. Antoine Henrot 1 Introduction 1.1 General introduction This collective book has the ambition to give an overview of recent results in spectral geometry and its links with shape optimization. The questions we are interested in are: – Does there exist a set which minimizes (or maximizes) the k-th eigenvalue of a given elliptic operator (mainly the Laplacian or the Schrödinger operator) with given boundary conditions (mainly Dirichlet, Robin or Steklov boundary condi- tions in this book) among sets of given volume? – If existence is proved, what can be said about the regularity of the optimal set? – Can one compute the optimal set? Can one give some geometric properties of it? – Is it possible to prove some stability results for the well-known isoperimetric in- equalities involving the first or the second eigenvalue, namely some quantitative isoperimetric inequalities for eigenvalues? – More generally, can one write some universal inequalities related to the eigenval- ues of elliptic operators? – For some more specific shapes like triangles, can one prove more precise bounds and observe some patterns of the eigenfunctions? A first survey on these kinds of extremum problems appeared in 2006 in the book "Extremum Problems for eigenvalues of elliptic operators", see [505]. It turns out that much progress has been made in the last ten years which leads us to think that an update and an extension would be valuable for a wide community who is interested in spectral theory and its links with geometry, calculus of variations, free boundary problems and shape optimization. 1.2 Content of the book We now describe in more detail the content of each chapter. For most of the nota- tion used here and in this book, we refer to the end of this Introduction chapter (Sec- tion 1.4). Antoine Henrot: Institut Élie Cartan, UMR 7502, Université de Lorraine, CNRS BP 70239, 54506 Vandoeuvre-lès-Nancy, France, E-mail: antoine.henrot@univ-lorraine.fr © 2017 Antoine Henrot This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
- 16. 2 | Antoine Henrot Existence results by D. Bucur In Chapter 2, existence of a minimizer for a problem like min{λk(Ω), Ω ⊂ Rd , |Ω| = c} (1.1) or more generally like min{F λ1(Ω), . . . λk(Ω) , Ω ⊂ Rd , |Ω| = c} (1.2) is studied. It is necessary to enlarge the class of admissible sets to quasi-open sets which is actually the largest, and more natural, class for which eigenvalues of the Dirichlet-Laplacian are well defined. A famous existence result due to G. Buttazzo and G. Dal Maso in [238] was previously available but with the supplementary assumption that the sets have to lie in some fixed bounded box D. A lot of effort has been done to remove this assumption and to get a general existence result. These efforts eventually led to the papers by D. Bucur [206] and A. Pratelli-D. Mazzoleni [700] where this open problem of existence was solved. These two papers use a completely different strategy. In [206], the notion of shape subsolution for the torsion energy was introduced, and it was proved that every such subsolution has to be bounded and has finite perimeter. A second argument, showed that minimizers for (1.1) are shape subsolutions, so they are bounded. The author finished the proof by using a concentration-compactness ar- gument, like in [219]. The approach of [700] is different: a surgery result proved that some parts (like long and tiny tentacles), can be cut out from every set such that, af- ter small modifications and rescaling, the new set has a diameter uniformly bounded and its first k eigenvalues are smaller. In this way, the existence problem in Rd can be reduced to the local case of Buttazzo and Dal Maso. In Chapter 2, the main ideas of the proof of the global existence result are provided, using a combination of the two methods: shape subsolutions and surgery. Moreover, it is proved that the optimal set is bounded and has finite perimeter. Regularity of optimal spectral domains by J. Lamboley and M. Pierre Once existence has been proved (for example for Problem (1.1)), an important and dif- ficult issue is to study the regularity of the optimal set Ω* . As we have seen, the exis- tence theorem provides a solution which is only quasi-open and this is a very weak regularity. Now it seems reasonable to expect much more regularity such as Lipschitz or C2 boundary, even analytic (at least in two dimensions). We would like to have this kind of regularity to be able to write optimality conditions thanks to the shape deriva- tive, see [510, chapter 5]. For computing such shape derivatives, a minimal regularity is required (e.g. C2 regularity if we want to consider the trace of the gradient on the boundary which occurs in the classical Hadamard’s formula). It turns out that the
- 17. Introduction | 3 main difficulty in this regularity issue is the first step: to be able to reach weak but consistent regularity, for example that the boundary of the optimal domain is locally a graph. Once this is done, it is often possible to use powerful tools developed in the theory of regularity for free boundary problems (e.g. by L. Caffarelli and co-authors) to reach the desired properties. Chapter 3 is devoted to these questions and presents the known results. The authors also point out many open problems which remain to be solved. In a first section, they consider the problem min{λ1(Ω), Ω ⊂ D, |Ω| = c} (1.3) where D is a given bounded domain. Of course, if D is large enough (or c small enough) in order that it contains the ball of volume c, this one is the solution and there is noth- ing to prove. Otherwise, the regularity of the optimal domain is studied in detail and is well understood. The fact that the state function (the first eigenfunction u1) is positive plays an important role in this analysis. Actually, the first step is to study the global regularity of u1 naturally extended by zero outside Ω* . For that purpose, the authors show that Problem (1.3) is actually equivalent to a penalized version min{λ1(Ω) + µ[|Ω| − c]+ , Ω ⊂ D} (1.4) (where [x]+ denotes the positive part of x) for µ large enough. It turns out that it is much more convenient to work with this penalized version, in particular it is easier to perform variations and exploit the minimality of the domain which is used to prove that the eigenfunction is globally Lipschitz continuous. This idea of the penalized ver- sion works as well for any eigenvalue λk but without a box constraint: Problem (1.1) is equivalent to min{λk(Ω) + µ|Ω|, Ω ⊂ Rd } (1.5) for a particular value of µ. Knowing that the state function is Lipschitz continuous is a first (important) step in the study of the regularity of the boundary of the optimal set, but obviously not sufficient. The next step is to prove that the gradient of the state function does not degenerate at the boundary, in order to be able to use some im- plicit function theorem to deduce regularity of the boundary itself. This is done in this chapter for λ1 where analyticity of the boundary is proved in dimension d = 2, while regularity of the reduced boundary is proved in higher dimension. Then, for Problem (1.3) the regularity of ∂Ω* up to the boundary of the box D is studied. At last, some problems with a convexity constraint are considered. The Robin problem by D. Bucur, P. Freitas and J. Kennedy While the two previous chapters deal with Dirichlet boundary conditions, the next one deals with Robin boundary conditions: ∂u ∂n + αu = 0 on ∂Ω. This is a very impor- tant case since it can be seen as a generalization (or interpolation) of the Dirichlet and
- 18. 4 | Antoine Henrot Neumann cases. Neumann boundary conditions correspond to α = 0 while Dirichlet ones correspond to α → +∞. Chapter 4 presents a very complete overview on qualita- tive properties for the Robin eigenvalues. Among many results shown in that chapter we can find a study of the curves α 7→ λk(α, Ω) together with a precise asymptotic expansion of λk(α, Ω) for both α → ±∞. Concerning isoperimetric inequalities, while the minimization problem for the first two eigenvalues is now well understood when α is positive, the corresponding maximization problems for negative α remains open. This is the last problem for which an isoperimetric inequality for the first eigenvalue of the Laplace operator has not yet been solved. Chapter 4 begins with a clear presen- tation of the minimization of λ1(α, Ω) for α 0 recalling the arguments used by M.H. Bossel and D. Daners. Then, this result is applied to solve the minimization problem for λ2(α, Ω) (result due to J. Kennedy). The situation for higher eigenvalues seems to be much more complex, since the optimizers are expected to depend on the bound- ary parameter α as suggested by numerical simulations. Nevertheless some properties are also given in that case. The case α 0 is still more intriguing. The long-standing conjecture that the ball should be the maximizer for every value of α has been very recently disproved by P. Freitas and D. Krejčiřík. More precisely, they proved that the disk is indeed the maximizer for small values of α (in dimension 2). But the ball can- not be the maximizer for large (negative) values of α since an annulus (for d = 2) or a spherical shell (for d ≥ 3) gives a larger value. This can be seen thanks to the pre- cise evaluation of the asymptotic expansion of λ1(α) when α → −∞ that was obtained previously. Spectral geometry of the Steklov problem by A. Girouard and I. Polterovich Chapter 5 presents an overview of the geometric properties of the Steklov eigenval- ues and eigenfunctions. One can consider two natural constraints for an isoperimet- ric inequality for the first non trivial Steklov eigenvalue. Under the volume constraint, the fact that the ball is the extremal domain has been proved by F. Brock. A perime- ter constraint appears to be more natural from the following viewpoint: the Steklov eigenvalues are the eigenvalues of the Dirichlet-to-Neumann operator, which is an op- erator defined on the boundary of the domain. In that case, Weinstock has proved that the disk is the maximizer in the class of simply-connected planar domains. However, this topological assumption cannot be removed, as can be seen from the example of an annulus. Moreover, for general Euclidean domains the question of existence of a maximizer remains open. Nevertheless, it is known that for simply connected planar domains, the k-th normalized Steklov eigenvalue is maximized in the limit by a dis- joint union of k − 1 identical disks for any k ≥ 2. Some geometric bounds are also obtained in higher dimensions, but they are more complicated, as they involve other geometric quantities, such as the isoperimetric ratio. Other interesting questions are also discussed in this chapter, in particular, isospectrality (Can one hear the shape
- 19. Introduction | 5 of a drum whose mass is concentrated on the boundary?). It is largely open, since the usual techniques applied in the Dirichlet case, such as the transplantation tech- nique, do not work for planar domains with the Steklov boundary condition. Another topic covered in the chapter is the study of nodal lines and nodal domains. Bounds for the multiplicity of the Steklov eigenvalues, as well as the asymptotic distribution of Steklov eigenvalues are also considered. It turns out that spectral asymptotics in the Steklov case strongly depend on the regularity of the domain. Triangles and other special domains by R.S. Laugesen and B. Siudeja Chapter 6 reports on known and conjectured spectral properties of the Laplacian on special domains, like triangles, rectangles or rhombi. Topics include sharp lower bounds and sharp upper bounds, as well as inverse problems, hot spots, and nodal do- mains. The authors consider both Dirichlet and Neumann boundary conditions (and sometimes mixed Dirichlet–Neumann conditions). This chapter begins with classical applications of symmetrization techniques for finding optimal domains, for example in the class of triangles. Then the method of unknown trial functions is presented and used to obtain sharp lower bounds for triangles, including a sharp Poincaré inequal- ity. The method consists in transplanting the (unknown) eigenfunction of an arbitrary triangle to yield trial functions for the (known) eigenvalues of certain equilateral and right triangles. The method of choosing clever trial functions allows one to get sharp upper bounds. For isosceles or right triangles, explicit expressions for the eigenval- ues are not available, but nevertheless, the authors are able to get some monotonic- ity formulas in these cases. The chapter also discusses the inverse problem Can one hear the shape of a triangular drum? and the positive answer by Durso and Grieser– Maronna. Lastly, qualitative properties of eigenfunctions and their nodal regions are investigated: – simplicity of the low eigenvalues λ2 and µ2 for triangles, – the hot spots conjecture for acute triangles, – the Courant-sharp property (which means that the number of nodal domains is exactly the rank of the eigenvalue), – the sign of the Neumann eigenfunctions on the boundary of the domain. Spectral inequalities in quantitative form by L. Brasco and G. de Philippis When an isoperimetric inequality, like Faber-Krahn inequality, is proved, a very natu- ral question is the stability issue. Namely: assume that a domain Ω has a first Dirichlet eigenvalue very close to the first Dirichlet eigenvalue of a ball of same volume, to what extent can we claim that Ω itself is close to a ball? Moreover, can we quantify it? This
- 20. 6 | Antoine Henrot kind of question has a long history for the classical (geometric) isoperimetric inequal- ity. Concerning the eigenvalues, it started in the 1990’s and enjoyed a renewed success during the ten last years particularly thanks to the Italian school. The aim of this chap- ter is to give a complete picture on recent results about quantitative improvements of sharp inequalities for eigenvalues of the Laplacian with all the classical boundary con- ditions. The authors begin by the case of the Faber-Krahn inequality. The distance to the ball (or more generally to the optimal domain) can be expressed in different ways. A popular choice consists in using the so-called Fraenkel asymmetry, which is a L1 distance between the characteristic functions: A(Ω) := inf |Ω∆B| |Ω| B ball such that |B| = |Ω| . (1.6) A discussion is undertaken so as to compare it to other measures of asymmetry. Then, the stability of the Szegő-Weinberger and Brock-Weinstock inequalities are treated. For each of these situations, the authors present the relevant stability result and then dis- cuss its sharpness, in particular the sharpness of the exponent on the Fraenkel asym- metry which occurs in the quantitative inequality and this is not always as simple as one may think. For the quantitative Faber-Krahn inequality, the sharp exponent is 2 and several weaker results were available in the literature before being able to get this exponent. The proof, whose main ideas are given here, consists in obtaining a quanti- tative estimate for the torsional rigidity by some selection principle. Some interesting applications of the quantitative Faber-Krahn inequality to estimates of the so called harmonic radius are also considered. Another Section is devoted to presenting the proofs of other spectral inequalities, involving the second Dirichlet eigenvalue λ2 as well, as the Hong-Krahn-Szego inequality for λ2 and the Ashbaugh-Benguria inequal- ity for the ratio λ2/λ1. For the Hong-Krahn-Szego inequality, it is important to notice that it is the first example of quantitative isoperimetric inequality for which the target set is no longer the ball, since the optimal domain is the disjoint union of two identical balls. Obviously, this requires a different version for the asymmetry adapted to this situation. Universal inequalities by M. Ashbaugh Inequalities involving eigenvalues are called universal when they hold in complete generality, requiring no hypotheses on the domain (other than that it is of dimension d). It contrasts with most of the isoperimetric inequalities of other chapters which are obtained with a volume (or perimeter) constraint. Famous examples of such inequal- ities (for the Dirichlet-Laplacian) are λ2 λ1 ≤ 1 + 4 d or more generally λm+1 − λm ≤ 4 md m X i=1 λi (1.7)
- 21. Introduction | 7 which were proved for (planar) domains by Payne, Pólya and Weinberger in [746] and which may be considered as the starting point of that study. Other more general re- sults have been obtained by Hile and Protter on the one hand and Yang on the other, and Chapter 8 shows in a very clear way all these results and the hierarchy between them. Let me also mention that the first inequality in (1.7) has been later improved as λ2/λ1 ≤ j2 d/2,1/j2 d/2−1,1 by the author and R. Benguria in [62], [64] giving here the sharp inequality. Chapter 8 presents in an unified way these universal inequalities by choosing the more general approach. All the proofs are given and the author often chooses to present simpler proofs or more general results than the original ones. In particular, several statements are given with the Laplacian replaced by a Schrödinger operator with both a scalar and vectorial potential. The authors considers whether it is possible to deduce from the previous universal inequalities some explicit bounds for a given eigenvalue, say λm+1. Moreover, we would like these bounds to be in good accordance with Weyl’s law. Several interesting results in that direction are presented. This chapter aims to put in perspective all of this material, by giving historical markers and references. Spectral optimization problems for Schrödinger operators by G. Buttazzo and B. Velichkov In this chapter Schrödinger operators of the form −∆+V(x) are considered, with Dirich- let boundary conditions on a bounded open set D ⊂ Rd . The question is now to find optimal potentials for some suitable optimization criteria. In general, the optimiza- tion problems studied here can be written as min{F(V), V ∈ V} where F is a suitable cost functional and V is a suitable class of admissible potentials. The case of spectral functionals min{Φ(λ(V)), V ∈ V}, where λ(V) is the spectrum of the Schrödinger op- erator, which will be assumed to be discrete, are also considered. Here, the interest is not only in potentials V which are bounded functions, but also in their natural exten- sion: the capacitary measures which are nonnegative Borel measures on D, possibly taking the value +∞ and vanishing on all sets of capacity zero. The class of capaci- tary measures is very large and contains both the cases of standard potentials V(x), in which µ = Vdx, as well as the case of classical domains, where we set µ = +∞DΩ which is the measure defined by µ(E) = ( 0 if cap(Ω E) = 0 +∞ if cap(Ω E) 0. (1.8) In that sense, this can be seen as a unified presentation of these two classical prob- lems. The authors are mainly interested in existence theorems. They first prove a very general existence result in the class of capacitary measures. Then they consider the more specific case of integrable potentials: V ∈ Lp (D) for which other general exis- tence theorems are proved. Some examples where the optimal potential can be ex-
- 22. 8 | Antoine Henrot plicitly determined are also presented. In the last section, they consider another class of admissible potentials, namely the function V(x) such that ´ D Ψ(V) dx ≤ 1 where Ψ satisfies some assumptions allowing possible large potentials. Nodal and spectral minimal partitions by V. Bonnaillie-Noël and B. Helffer This chapter is devoted to the analysis of minimal partitions and their relations with the nodal domains of eigenfunctions. Let Ω be a fixed domain, then a k-partition D of Ω is a family of k disjoint sub-domains of Ω: D1, D2, . . . , Dk. It is a natural and popular question to ask what possible k-partition D minimizes Λ(D) := max λ1(D1), . . . , λ1(Dk) , (1.9) where λ1(Di) denotes the first Dirichlet eigenvalue of the sub-domain Di? In particu- lar, is this minimal partition related to the nodal domains of a given eigenfunction, for example the k-th eigenfunction associated to λk(Ω). As the reader will discover, this is always the case for k = 2, but this is true for higher values of k if and only if λk(Ω) is Courant-sharp which means that it has a corresponding eigenfunction with exactly k nodal domains, saturating in this way the famous Courant nodal Theorem. This minimal partition problem has strong links with models in mathematical ecology where the sub-domains represent the strong competition limit of segregating species in population dynamics. A complete overview of what is known about Courant-sharp eigenvalues is presented here. The famous Pleijel’s Theorem states that there are only a finite number of such eigenvalues and even gives an upper bound of the number of nodal domains. For some particular domains (not many), it is possible to give explic- itly the eigenvalues which are Courant-sharp. As explained, this occurs in particular the case for thin domains, square, some rectangles, some torus, disk and circular sec- tors. Then, the minimal partitions are studied. Existence and regularity of minimal partitions is stated. The case of 3-partitions and their possible topologies are investi- gated in more detail. Some explicit results and conjectures, supported by numerical simulations are given. A generalization to p-minimal k-partitions where the `∞ norm defining Λ in (1.9) is replaced by the `p norm is also considered. Then the authors in- troduce the Aharonov-Bohm operators. It turns out that minimal partitions can be rec- ognized as nodal partitions of eigenfunctions of these operators. This gives interesting necessary conditions for candidates to be minimal partitions. At last the asymptotic behavior of minimal partitions when k → +∞ is discussed. In particular the hexagonal conjecture and some other qualitative properties are presented.
- 23. Introduction | 9 Numerical results for spectral optimization problems by P. Antunes and E. Oudet This chapter is devoted to numerical methods which have been introduced to solve the previous problems. In the first two sections two of these approaches which have been successful in recent years on spectral problems are explained. The first one consists in introducing some global optimization tools to provide a good initial guess of the optimal profile. This step does not require any topological information on the set but is restricted to a small class of shapes. Then the method of fundamental solutions (to compute the eigenvalues) is described. It allows, in a second stage, the identification and precise evaluation of shapes which are locally optimal. A constant preoccupation is to decrease the complexity of the optimization problem by introducing a reduction of the number of parameters which still allows a precise computation of the cost func- tion. For example, the parametrization of the boundaries of the open sets as level set functions, for example level sets of truncated Fourier series can be very efficient. The chapter ends with a presentation of the best domains obtained numerically for both Dirichlet and Neumann eigenvalues λk and µk (for k = 1 to 10 or 15) and some con- jectures inspired by these numerical results. 1.3 Balls and union of balls One of the most important topics discussed in this book is the determination of which domain minimizes or maximizes a given eigenvalue. For low eigenvalues, actually the two first eigenvalues, and for most boundary conditions, the optimal domains are known and it turns out that they are the same: the ball for the first eigenvalue and the union of two identical balls for the second. These results are recalled in different chapters of this book, but let us sum up it here. First eigenvalue-Dirichlet The ball minimizes λ1(Ω) among sets of given volume (Faber-Krahn inequality), see [377] and [603]. First (non-trivial) eigenvalue-Neumann The ball maximizes µ2(Ω) among sets of given volume (Szegő-Weinberger inequality) , see [838] for Lipschitz simply con- nected planar domains and [871] for the general case. First eigenvalue-Robin (α 0) The ball minimizes λ1(Ω, α) among sets of given vol- ume (Bossel-Daners inequality), see [169] for the two-dimensional case and [319] for the general case. First eigenvalue-Steklov The ball maximizes σ2(Ω) among sets of given volume (Brock-Weinstock inequality), see [873] for the two-dimensional case and [192] for the general case.
- 24. 10 | Antoine Henrot Fig. 1.1. Left: the disk minimizes the first eigenvalue (Dirichlet or Robin) and maximizes the first non trivial eigenvalue (Neumann or Steklov). Right: two disks minimizes the second eigenvalue (Dirichlet or Robin) and maximizes the second non trivial eigenvalue (Neumann or Steklov) Second eigenvalue-Dirichlet The union of two identical balls minimizes λ2(Ω) among sets of given volume (Hong-Krahn-Szego inequality), see [604], [534]. Second (non-trivial) eigenvalue-Neumann The union of two identical disks maxi- mizes µ3(Ω) among simply connected bounded planar domains of given volume, see [428]. Second eigenvalue-Robin (α 0) The union of two identical balls minimizes λ1(Ω, α) among sets of given volume, see [586] and Theorem 4.36 in Chapter 4. Second eigenvalue-Steklov The union of two identical disks maximizes σ3(Ω) among simply connected bounded planar domains of given volume, see [513], [430] and Chapter 5. In view of the previous results, it is a natural question to ask whether there are other eigenvalues for which balls or union of balls could be the optimal domain. For Dirich- let eigenvalues, this question has been recently investigated in the PhD thesis of A. Berger, see [137]. She proves that for d = 2, only λ1 and λ3 can be minimized by the disk (it is still a conjecture for λ3). Moreover, only λ2 and λ4 can be minimized by union of disks (it is still a conjecture for λ4). Let us finish this section by giving the eigenvalues of the ball. In dimension 2, the eigenvalues of the disk BR of radius R for the Laplacian with Dirichlet boundary conditions and the corresponding eigenfunctions (not normalized) are given by λ0,k = j2 0,k R2 , k ≥ 1, u0,k(r, θ) = J0(j0,kr/R), k ≥ 1, λn,k = j2 n,k R2 , n, k ≥ 1, double eigenvalue un,k(r, θ) = ( Jn(jn,kr/R) cos nθ Jn(jn,kr/R) sin nθ , n, k ≥ 1, (1.10) where jn,k is the k-th zero of the Bessel function Jn. For the Laplacian with Neumann boundary conditions, the eigenvalues and eigen-
- 25. Introduction | 11 functions of the disk BR are: µ0,k = j′ 0,k 2 R2 , k ≥ 1, v0,k(r, θ) = J0(j′ 0,kr/R), k ≥ 1, µn,k = j′ n,k 2 R2 , n, k ≥ 1, double eigenvalue vn,k(r, θ) = ( Jn(j′ n,kr/R) cos nθ Jn(j′ n,kr/R) sin nθ , n, k ≥ 1, (1.11) where j′ n,k is the k-th zero of J′ n (the derivative of the Bessel function Jn). In dimension three, for the ball BR, the eigenvalues and eigenfunctions of the Dirichlet-Laplacian are given by λn,k = j2 n+ 1 2 ,k /R2 , n ∈ N, k ∈ N which is of multi- plicity 2n + 1 and is associated to the eigenfunctions vn,k(r, θ, ϕ) = Jn+ 1 2 j n+ 1 2 ,k R r ! √ r P0 n(cos θ), Jn+ 1 2 j n+ 1 2 ,k R r ! √ r P1 n(cos θ) cos ϕ, Jn+ 1 2 j n+ 1 2 ,k R r ! √ r P1 n(cos θ) sin ϕ, . . . Jn+ 1 2 j n+ 1 2 ,k R r ! √ r Pn n(cos θ) cos(nϕ), Jn+ 1 2 j n+ 1 2 ,k R r ! √ r Pn n(cos θ) sin(nϕ) (1.12) where Pq n denote the associated Legendre polynomial, see [4]. Similar formulae hold for the Neumann eigenvalues and eigenfunctions where the eigenvalues are the roots of some transcendental equations involving Bessel functions. In higher dimen- sion d, the eigenvalues of the ball BR still involve the zeros of the Bessel functions Jd/2−1, Jd/2, . . .. For example λ1(BR) = j2 d/2−1,1 R2 λ2(BR) = λ3(BR) = . . . = λN+1(BR) = j2 d/2,1 R2 (1.13) while the eigenfunctions combine Bessel functions for the radial part and spherical harmonics for the angular part, see [292] 1.4 Notation Ω is an open set (or a quasi-open set, see Chapter 2) in Rd . We will denote by H1 (Ω) the classical Sobolev space:
- 26. 12 | Antoine Henrot H1 (Ω) = u ∈ L2 (Ω), such that ∂u ∂xi ∈ L2 (Ω), i = 1, . . . , d and H1 0(Ω) is defined as the closure in H1 (Ω) of C∞ functions with compact support in Ω. Eigenvalues and eigenfunctions of the Laplace-Dirichlet operator. We will denote by λk(Ω), k ≥ 1 (or more simply λk when the context makes the domain clear) the k-th eigenvalue of the Laplacian with Dirichlet boundary conditions, counted with multiplicity: ( −∆u = λ u in Ω, u = 0 on ∂Ω. (1.14) The corresponding eigenfunction is usually normalized by ´ Ω u2 dx = 1. Eigenvalues and eigenfunctions of the Laplace-Neumann operator. We will de- note by µk(Ω), k ≥ 1 (or more simply µk when the context makes the domain clear) the k-th eigenvalue of the Laplacian with Neumann boundary conditions, counted with multiplicity: ( −∆u = µ u in Ω, ∂u ∂n = 0 on ∂Ω. (1.15) Therefore, by convention, µ1(Ω) = 0. The corresponding eigenfunction is usually nor- malized by ´ Ω u2 dx = 1. Eigenvalues and eigenfunctions of the Laplace-Robin operator. Let α a real num- ber, we will denote by λk(Ω, α) (or more simply λk(α) or λk when no confusion can occur) the k-th eigenvalue of the Laplacian with Robin boundary conditions, counted with multiplicity: ( −∆u = λu in Ω, ∂u ∂n + αu = 0 on ∂Ω. (1.16) The corresponding eigenfunction is usually normalized by ´ Ω u2 dx = 1. Eigenvalues and eigenfunctions of the Laplace-Steklov operator. We will denote by σk(Ω) (or more simply σk when the context makes the domain clear) the k-th eigen- value of the Laplacian with Steklov boundary conditions, counted with multiplicity: ( ∆u = 0 in Ω, ∂u ∂n = σu on ∂Ω. (1.17) Therefore, by convention, σ1(Ω) = 0. Here Ω can be a compact Riemannian manifold of dimension n ≥ 2 and in that case ∆ is the Laplace-Beltrami operator. The corre- sponding eigenfunction can be normalized either by ´ Ω u2 dx = 1 or more frequently by ´ ∂Ω u2 dσ = 1.
- 27. Dorin Bucur 2 Existence results 2.1 Setting the problem In this chapter, we denote by d the dimension of the space. Let d ≥ 2 and Ω ⊆ Rd be an open set of finite measure. Then the spectrum of the Laplace operator with Dirichlet boundary conditions of Ω consists only on eigenvalues which can be ordered (count- ing the multiplicity) 0 λ1(Ω) ≤ λ2(Ω) ≤ · · · → +∞. For every k ∈ N there exist non-zero functions u (eigenfunctions) that saisfy the equa- tion ( −∆u = λk(Ω)u in Ω u = 0 on ∂Ω. If Ω is smooth, then the function u ∈ C2 (Ω)∩ C(Ω) satisfies the equation in a classical sense. If Ω is just an open set without any regularity, then u satisfies the equation in the following weak sense u ∈ H1 0(Ω), ∀v ∈ H1 0(Ω), ˆ Ω ∇u∇vdx = λk(Ω) ˆ Ω uvdx. Let c 0, k ∈ N be given, and F : Rk → R. The generic spectral optimization problems we discuss in this chapter is min n F λ1(Ω), .., λk(Ω) : Ω ⊂ Rd , Ω open, |Ω| = c o . (2.1) Some particular problems have been studied intensively in the last century. We refer the reader to [505] for a recent survey of the topic. Here is a short list of results. – The Faber-Krahn inequality asserts that the solution of min n λ1(Ω) : Ω ⊂ Rd , Ω open, |Ω| = c o (2.2) is the ball of volume c. – The solution of min n λ2(Ω) : Ω ⊂ Rd , Ω open, |Ω| = c o (2.3) consists of two equal and disjoint balls of volume c 2 (Krahn-Szegő). Dorin Bucur: Institut Universitaire de France, Laboratoire de Mathématiques, CNRS UMR 5127, Université de Savoie, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France, E-mail: dorin.bucur@univ-savoie.fr © 2017 Dorin Bucur This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
- 28. 14 | Dorin Bucur – Ashbaugh and Benguria proved in [64] that the solution of max n λ2(Ω) λ1(Ω) : Ω ⊂ Rd , Ω open and of finite measure o (2.4) is the ball. An intriguing question is to find the solution of min n λk(Ω) : Ω ⊂ Rd , Ω open, |Ω| = c o (2.5) for every k ∈ N. Unfortunately, starting with k ≥ 3 very few answers are available. In two dimensions, for k = 3 it is conjectured that the minimizer is the disc, while in dimension 3 it has been observed numerically by Oudet that the minimizer is not the ball, cf Chapter 11. Wolf and Keller proved that for k = 13, in R2 , the minimizer is not a union of discs, but again Oudet [737] numerically observed that for k = 5 to 15 the minimizer is not the disc. In [137], it was rigorously proved by Berger, that for any k ≥ 5 in R2 , the ball cannot be minimizer. Several computations were carried out ([42, 737]) providing evidence that the op- timal shapes are close to those presented in Chapter 11. At this point, when no analytical solutions of those problems can be expected, the question is of a qualitative nature. One would like to prove that problem (2.1), or more precisely problem (2.5), has a solution and to gather some information about it. Does the optimal set have finite perimeter ? Is it bounded ? Is its boundary smooth ? Does it have any symmetry ? Is it convex ? Is it the ball ? Problem (2.1) may have or not a solution (for general functions F, the complete answer is not known), but a negative answer may have at least three meanings: – a solution of problem (2.1) does not exist (i.e. in the class of open sets), but there exists a solution provided the family of open sets is enlarged to a class of Borel subsets of Rd where the eigenvalue problem is still well posed (i.e. the family of quasi-open sets, see Section 2.2 below). This issue is very similar to a classical ex- istence result, only that the solution is a quasi-open set. In fact the class of quasi- open sets is the largest class of Borel subsets of Rd , where the Dirichlet Laplacian is well defined and inherits a strong maximum principle. – a solution of problem (2.1) does not exist, even if the class of sets is enlarged, but there exists a solution in a larger class of relaxed objects where the eigenvalue problem is well posed. This class consists of positive Borel measures, absolutely continuous with respect to capacity (see Section 2.2 below for definition and prop- erties of capacity). Roughly speaking, those measures are limits of sequences of open sets in some suitable sense, and account for the asymptotic behavior of the oscillating boundaries in the sense of capacities (see Remark 2.5 below). – a solution of problem (2.1) does not exist, in the sense that the infimum is not attained by any geometrical object.
- 29. 2 Existence results | 15 In this chapter, we shall analyze the existence of a solution for problem (2.1), and we shall prove that it indeed exists (in the enlarged class of quasi-open sets), provided some assumptions are satisfied by the functional F. Of course, one expects to have smooth open sets as minimizers, at least for problem (2.5), but this has not yet been proved in general. The question of proving existence in the class of open sets is in fact a regularity problem for which we refer the reader to Chapter 3. Some qualitative properties will be proved in this chapter, e.g. the boundedness of the optimal sets and the fact that they have a finite perimeter, since they play a crucial role in the existence question. 2.2 The spectrum of the Dirichlet-Laplacian on open and quasi-open sets Since the existence question requires us to work in a class of sets larger than the class of open sets, in this section we recall basic facts about capacity and quasi-open sets. We also list some properties of the eigenvalues of the Dirichlet-Laplacian and of the eigenfunctions. Capacity and quasi-open sets. Let E ⊆ Rd . The capacity of E is defined by cap(E) = inf n ˆ |∇u|2 + |u|2 dx, u ∈ UE o where UE is the class of all functions u ∈ H1 (Rd ) such that u ≥ 1 almost everywhere (shortly a.e.) in an open neighborhood of E. A property p(x) is said to hold quasi everywhere on E (shortly q.e. on E) if the set of all points x ∈ E for which the property p(x) does not hold has capacity zero. A set Ω ⊆ Rd is called quasi-open if for every ϵ 0 there exists an open set Uϵ such that Ω ∪ Uϵ is open and cap(Uϵ) ϵ. Clearly, every open set is quasi-open. A function u : Rd 7→ R is said to be quasi-continuous if for all ϵ 0 there exists an open set Uϵ with cap(Uϵ) ϵ such that u|Uc ϵ is continuous (see [472]). Every function u ∈ H1 (Rd ) has a quasi-continuous representative, ũ, such that ũ(x) = u(x) a.e. This representative is unique up to a set of zero capacity and can be computed by ũ(x) = lim r→0 ´ Br(x) u(y)dy |Br(x)| , q.e. x ∈ Rd . The limit above exists quasi everywhere. In particular, the level set {ũ 0} is a quasi- open set. From now on, every time we speak about the pointwise behavior of a Sobolev function, we refer to a quasi-continuous representative. The Sobolev spaces. If Ω ⊆ Rd is an open set, the Sobolev space H1 0(Ω) is defined as clH1(Rd)C∞ 0 (Ω), the closure of the space of C∞ functions with compact support in Ω, in
- 30. 16 | Dorin Bucur the H1 -norm. For a quasi-open set Ω ⊆ Rd , the Sobolev space H1 0(Ω) is defined as a subspace of H1 (Rd ) by: H1 0(Ω) = {u ∈ H1 (Rd ) : u = 0 q.e. on Rd Ω}. If Ω is open, the space H1 0(Ω) defined above coincides with the usual Sobolev space (see [472]). From this perspective, for every open or quasi-open set, H1 0(Ω) is a subspace of H1 (Rd ), as long as every function of H1 0(Ω) is understood as being extended by 0 on Rd Ω. If Ω ⊆ Rd is a quasi-open set of finite measure (not necessarily bounded), the injection H1 0(Ω) ,→ L2 (Ω) is compact. The spectrum of the Dirichlet-Laplacian. For every quasi-open set of finite measure Ω ⊆ Rd , we introduce the resolvent operator RΩ : L2 (Rd ) → L2 (Rd ), by RΩ(f) = u, where u solves the equation ( −∆u = f in Ω u ∈ H1 0(Ω) in the weak sense ∀ϕ ∈ H1 0(Ω) ˆ Ω ∇u∇ϕdx = ˆ Ω fϕdx. (2.6) RΩ is a compact, self-adjoint, positive operator having a sequence of eigenvalues converging to 0. The inverses of its eigenvalues are the eigenvalues of the Dirichlet- Laplacian on Ω and are denoted (multiplicity being counted) by 0 λ1(Ω) ≤ λ2(Ω) ≤ · · · ≤ λk(Ω) ≤ · · · → +∞. These values can be defined by the min-max formula λk(Ω) = min S∈Sk max u∈S{0} ´ Ω |∇u|2 dx ´ Ω u2dx , (2.7) where Sk stands for the family of all subspaces of dimension k in H1 0(Ω). A function u ∈ H1 0(Ω) for which equality holds, is called an eigenfunction and satisfies the equation ( −∆u = λk(Ω)u in Ω u = 0 on ∂Ω in the weak sense ∀ϕ ∈ H1 0(Ω) ˆ Ω ∇u∇ϕdx = λk(Ω) ˆ Ω uϕdx. The system of L2 -normalized eigenfunctions is a Hilbert basis of H1 0(Ω). We list below some properties of the eigenvalues. Let Ω, Ω1, Ω2 ⊆ Rd be quasi- open sets of finite measure.
- 31. 2 Existence results | 17 – (Rescaling) ∀t 0, ∀k ∈ N, λk(tΩ) = 1 t2 λk(Ω). – (Spectrum of the union) If Ω1, Ω2 are disjoint, then the eigenvalues of Ω1 ∪ Ω2 are the union of the sets of eigenvalues of Ω1 and Ω2 with multiplicities being counted. – (Monotonicity) Assume that Ω1 ⊆ Ω2. Then ∀k ∈ N, λk(Ω2) ≤ λk(Ω1). – (Control of the variation) 1 λk(Ω1) − 1 λk(Ω2) ≤ kRΩ1 − RΩ2 kL(L2(Rd)). Moreover, if Ω1 ⊆ Ω2 (see Bucur [206]), for every k ∈ N 1 λk(Ω1) − 1 λk(Ω2) ≤ 4k2 e1/4π λk(Ω2)d/2 (E(Ω1) − E(Ω2)). (2.8) where E(Ω) := min u∈H1 0(Ω) 1 2 ˆ Rd |∇u|2 dx − ˆ Rd udx. (2.9) is the torsion energy. The unique function which minimizes E(Ω) is called the tor- sion function and is denoted wΩ and satisfies in a weak sense −∆wΩ = 1 in Ω, wΩ ∈ H1 0(Ω). The torsion function plays a key role in understanding the behavior of the spec- trum of the Dirichlet Laplacian for small geometric domain perturbations. Assume that λk(Ω2) = K and Ω1 ⊆ Ω2 such that E(Ω1) − E(Ω2) = 1 2 ˆ Ω2 (wΩ2 − wΩ1 )dx ≤ 1 8k2e1/4πKd/2+1 . (2.10) Then we get a control on the magnitude of λk(Ω1). Precisely, if (2.10) holds, we get λk(Ω1) ≤ 2λk(Ω2). (2.11) As ˆ Ω2 (wΩ2 −wΩ1 )dx becomes smaller, the eigenvalues λk(Ω1) and λk(Ω2) become closer. Roughly speaking, this property asserts that the variation of the eigenvalues for inner perturbations of a quasi-open set is controlled by the variation of the L1 - norm of the torsion functions (see Section 2.5). – (Control by the torsion function, see Van den Berg [860]) 1 λ1(Ω) ≤ kwΩk∞ ≤ 4 + 3d log 2 λ1(Ω) . (2.12)
- 32. 18 | Dorin Bucur – (Ratio of eigenvalues) For all k ∈ N there exists a constant Mk, depending only on k and the dimension d, such that (see for instance [57]) 1 ≤ λk(Ω) λ1(Ω) ≤ Mk. (2.13) – (L∞ -bound of the eigenfunctions) If uk is an L2 -normalized eigenfunction of λk(Ω), then kukk∞ ≤ Cdλk(Ω) d 4 . (2.14) 2.3 Existence results: bounded design region Let D ⊆ Rd be a bounded open set. In this section we shall prove an existence result for a local version of problem (2.1), i.e. min F λ1(Ω), .., λk(Ω) : Ω ⊂ D, Ω quasi-open, |Ω| = c (2.15) Theorem 2.1. (Buttazzo-Dal Maso) Let F : Rk → R be non-decreasing in each vari- able and lower semicontinuous. Then problem (2.15) has at least one solution. The first proof of this theorem, given by Buttazzo and Dal Maso in [238], involved quite technical results describing the so called relaxation phenomenon and covered a more general situation than just a functional depending on the eigenvalues. We give below a direct proof, which does not require the knowledge of the relaxed problem or prop- erties of the weak gamma convergence (see Remark 2.6), since we are concerned only with functionals depending on eigenvalues. A series of remarks at the end of the proof will explain the necessity of the mono- tonicity hypothesis on F and the boundedness of the design region D. Proof. Assume that (Ωn)n is a minimizing sequence for problem (2.15) and that u1 n, . . . , uk n are L2 -normalized eigenfunctions corresponding to λ1(Ωn), . . . , λk(Ωn), two by two orthogonal in L2 (D). We can assume that (λk(Ωn))n is a bounded sequence, otherwise existence occurs trivially as a consequence of the monotonicity of F (from some rank on, F will be constant on the sets Ωn). After extracting a subsequence we can assume that (ui n)n converges weakly in H1 0(D), strongly in L2 (D) and pointwise a.e. to a function ui ∈ H1 0(D), for i = 1, . . . , k. We can also assume that ui are quasi- continuous, so that defining Ω := k [ i=1 {ui 6= 0}, we built a quasi-open set Ω ⊆ D such that ui ∈ H1 0(Ω). From the pointwise a.e. con- vergence, we get 1{ui 6 = 0} ≤ lim inf n→∞ 1Ωn a.e.
- 33. 2 Existence results | 19 so that 1Ω ≤ lim infn→∞ 1Ωn a.e. This implies |Ω| ≤ c. On the other hand λi(Ω) ≤ lim inf n→∞ λi(Ωn), ∀i = 1, . . . , k. Indeed, this is a consequence of the definition of the eigenvalues. Let us denote Si = span{u1 , . . . , ui }. We have, for some (αj)i j=1 ∈ Ri λi(Ω) ≤ max u∈Si ´ Ω |∇u|2 dx ´ Ω |u|2dx = ´ Ω | Pi j=1 αj∇uj |2 dx ´ Ω | Pi j=1 αjuj|2dx . Hence λi(Ω) ≤ lim inf n→∞ ´ Ωn | Pi j=1 αj∇uj n|2 dx ´ Ωn | Pi j=1 αjuj n|2dx ≤ lim inf n→∞ λi(Ωn), the last inequality being a consequence of the min-max formula on Ωn. Using the lower semincontinuity and the monotonicity of F, this gives F(λ1(Ω), . . . , λk(Ω)) ≤ lim inf n→∞ F(λ1(Ωn), . . . , λk(Ωn)). If |Ω| c, then adding an open set U to Ω such that U ⊆ D and |Ω ∪ U| = c, we get a solution for (2.15). This is again a consequence of the monotonicity of F and of the eigenvalues on inclusions of sets. Remark 2.2. The monotonicity of the functional F is crucial. From a technical point of view, this hypothesis is used above in the construction of the optimal set: the limit set Ω may have a measure strictly lower than c, and so by enlarging it, we get the solution. When enlarging the set Ω, the eigenvalues do not increase! The functional F is tailored to behave well during this procedure. Nevertheless, there are situations in which existence of a solution holds for functionals F that are not required to satisfy this property. This is the case of functionals depending only on λ1(Ω) and λ2(Ω) (see [207, Section 6.4]). It is not known whether a general existence result as Theorem 2.1 may hold if the monotonicity assumption on F is dropped and replaced by a weaker hypothesis. Remark 2.3. The boundedness of the design region D plays an important role for com- pactness, in the construction of the optimal set Ω. The only fact we used in the proof was that H1 0(D) is compactly embedded in L2 (D), so Theorem 2.1 holds with this hy- pothesis, instead the stricter hypothesis D bounded. Nevertheless, if D is not bounded, the existence of a solution may fail. For example, in R2 we consider D = +∞ [ i=3 B 1 2 − 1 i (i, 0), c = π 4 , F(Ω) = λ1(Ω), where Br(x) is the ball centered at x of radius r.
- 34. 20 | Dorin Bucur Then the infimum of the functional is λ1(B 1 2 ), which is not attained. For general unbounded design regions D, Theorem 2.1 may not apply. Neverthe- less, in the particular, and very important, case D = Rd , the existence result holds. This issue is discussed in the next section. Remark 2.4. Depending on F, the optimal set Ω may satisfy some regularity proper- ties. We refer to Theorem 2.10 in the next section and to Chapter 3. Remark 2.5. Given an arbitrary sequence of quasi-open subsets of D, the full behavior of the spectrum for at least one subsequence is completely understood. In fact, it was proved (see Dal Maso and Mosco [314]), that there exists a subsequence (still denoted using the same index) and a positive Borel measure µ, absolutely continuous with respect to the capacity, such that the sequence of resolvent operators RΩn : L2 (D) → L2 (D) converges in the operator norm on L2 (D) to Rµ, defined by Rµ(f) = uµ,f ( −∆uµ,f + µuµ,f = f in D uµ,f ∈ H1 0(D) ∩ L2 (D, µ) in the sense ∀ϕ ∈ H1 0(D) ∩ L2 (D, µ) ˆ D ∇uµ,f ∇ϕdx + ˆ D uµ,f ϕdµ = ˆ D fϕdx. (2.16) The function uµ,f is also the unique minimizer in H1 0(D) ∩ L2 (D, µ) of the functional u 7→ 1 2 ˆ D |∇u|2 dx + 1 2 ˆ D u2 dµ − ˆ D fudx. Then, λk(µ) are the inverses of the eigenvalues of the positive, self-adjoint and compact operator Rµ and are defined via the min-max formula λk(µ) = min S∈Sk max u∈S{0} ´ D |∇u|2 dx + ´ D u2 dµ ´ D u2dx , (2.17) where Sk stands for the family of all subspaces of dimension k in H1 0(D) ∩ L2 (D, µ). As a consequence of the convergence of the resolvent operators, for every k ∈ N, λk(Ωn) → λk(µ). Of course, from the point of view of the existence theorem 2.1, the measure µ does not provide a solution. The original proof of Buttazzo-Dal Maso (done for a larger class of functionals) consisted in replacing the measure µ, obtained from a minimizing sequence, by a quasi-open set which is, roughly speaking, the union of all sets of µ-finite measure. This set was proved to be optimal, thanks to the monotonicity of F. Remark 2.6. There is a second proof of Theorem 2.1, given in [210], which is based on the so called weak gamma convergence. It is said that Ωn weakly gamma converges to Ω if (wΩn )n converges weakly in H1 0(D) to some function w, and Ω = {w 0}. This
- 35. 2 Existence results | 21 convergence is compact in the family of quasi-open subsets of D, and the Lebesgue measure is lower semicontinuous. The key consequence of this convergence is the fol- lowing property. – Assume Ωn weakly gamma converges to Ω. For all sequences (unk )k, such that unk ∈ H1 0(Ωnk ) and unk converges weakly in H1 0(D) to some function u, then u has to belong to H1 0(Ω). Following the proof of Theorem 2.1, this property implies immediately that ∀k ∈ N, λk(Ω) ≤ lim inf n→∞ λk(Ωn). As a consequence, existence in Theorem 2.1 comes by a compactness-semicontinuity argument. Remark 2.7. More importantly, the argument of the previous remark works for ev- ery k ∈ N, since the set Ω is built from the torsion functions and not from the first eigenfunctions. As a consequence, the existence result can be extended to function- als depending on the full spectrum. So, if F : RN → R is lower semicontinuous in a suitable sense and non decreasing in each variable, then the existence result proved in Theorem 2.1 could apply. An example would be Ω 7→ ∞ X k=1 e−λk(Ω) −1 . 2.4 Global existence results: the design region is Rd In this section we deal with the problem min n F λ1(Ω), .., λk(Ω) : Ω quasi-open, Ω ⊆ Rd , |Ω| = c o (2.18) The passage from a bounded design region D to Rd is not trivial. In fact, the com- pact embedding of H1 0(D) in L2 (D), which played a crucial role in the proof, fails to be true in Rd . For example, in the proof of Theorem 2.1, if D = Rd , the limit functions ui may all be equal to zero. This occurs, for instance, if the sets Ωn have distances to the origin tending towards +∞. From this perspective, a first attempt to solve the existence problem in Rd was done in [219] and was based on the concentration compactness principle of Pierre- Louis Lions. The following result holds (see [204, 207]). Theorem 2.8. (Bucur) Let (Ωn)n be a sequence of quasi-open sets of Rd of measure equal to c. One of the following situations holds.
- 36. 22 | Dorin Bucur Compactness: There exists a subsequence (Ωnk )k∈N, a sequence of vectors (yk)k∈N ⊆ Rd and a positive Borel measure µ, vanishing on sets of zero capacity, such that kRyk+Ωnk − RµkL(L2(Rd)) → 0 as k → +∞ and S k∈N H1 0(Ωnk ) is collectively compactly embedded in L2 (Rd ). Dichotomy: There exists a subsequence (Ωnk )k∈N and a sequence of subsets Ω̃k ⊆ Ωnk , such that kRΩnk − RΩ̃k kL(L2(Rd)) → 0, and Ω̃k = Ω1 k ∪ Ω2 k with d(Ω1 k , Ω2 k) → ∞ and lim inf n→∞ |Ωi k| 0 for i = 1, 2. In [219], this result was used to prove the existence of a minimizer for λ3. Following Theorem 2.8, a minimizing sequence (Ωn)n can be either in the compactness case, in which we get existence, or in the dichotomy case, in which case, the minimizing sequence can be chosen such that it consists of disconnected sets. In this situation, the problem is reduced to finding the minimizers of λ1 and λ2 which are known. In order to use this argument to prove the existence of a minimizer for λ4, it would be enough to prove that the minimizer for λ3 is, for instance, bounded. In this case, the dichotomy would lead to a combination of a minimizer of λ3 and a ball. If the min- imizer of λ3 (that we know its existence) was not bounded, then the union of the min- imizer and a ball may always have a non-trivial intersection. This would be the case if the minimizer of λ3 was a dense set in Rd . Of course, this situation is not expected, but to exclude it one has to understand some qualitative properties of the minimizers. The global existence result in Rd , was proved independently in [206] and [700], by completely different methods. In [206], the notion of shape subsolution for the torsion energy (see the next section) was introduced, and it was proved that every such sub- solution has to be a bounded set with finite perimeter. A second argument showed that minimizers for (2.18) are shape subsolutions, so they are bounded, hence the concentration-compactness theorem 2.8 can be used. The proof in [700] used a surgery result which states that from every set with a diameter large enough, some parts can be cut out such that after small modifications and rescaling, the new set has a diameter below some treshold and not larger (low) eigenvalues. In this way, replacing the minimizing sequence, the existence problem in Rd was reduced to the local case of Buttazzo and Dal Maso. In the next section, we shall give the main ideas of the proof of the global existence result, using a combination of the two methods: shape subsolutions and surgery. The following result was proved in [221], as an extension of the surgery result of [700] and using the subsolution method of [206]. Lemma 2.9. (surgery) For every K, c 0, there exists D, C 0 depending only on K, c and the dimension d such that for every quasi-open set Ω ⊂ Rd with |Ω| = c there exists a quasi-open set Ω̃ with |Ω̃| = c, diam (Ω̃) ≤ D, Per(Ω̃) ≤ C and, if for some k ∈ N it
- 37. 2 Existence results | 23 holds λk(Ω) ≤ K, then ∀1 ≤ i ≤ k, λi(Ω̃) ≤ λi(Ω). (2.19) Moreover, if Per(Ω) C the inequalities (2.19) are strict. We shall give the lines of the proof of this lemma in the next section. Here is the main consequence. Theorem 2.10. (Bucur, Mazzoleni, Pratelli) Let F : Rk → R be non-decreasing in each variable and lower semicontinuous. Then problem (2.18) has at least one solution. If F is strictly increasing in at least one variable, then every solution of (2.18) is a bounded set with finite perimeter. Proof. Let (Ωn)n be a minimizing sequence for (2.18). One can choose Ωn such that λk(Ωn) ≤ K for all k and some suitable value K. Otherwise, the minimum of F would be formally achieved at (+∞, . . . , +∞), which from the monotonicity assumption implies that F has to be constant. As a consequence, we can use the surgery Lemma 2.10 and find a new sequence (Ω̃n) which has a uniformly bounded diameter and satisfies the measure constraint. Since ∀1 ≤ i ≤ k, λi(Ω̃) ≤ λi(Ω), the monotonicity of F implies that this new sequence is also minimizing. At this point, we can use the Buttazzo-Dal Maso Theorem 2.1, up to possible translations of all Ω̃ in the ball centered at the origin of radius D, and get the existence of an optimal set. Assume now that Ω is optimal and F is strictly increasing in at least one variable. If Per(Ω) C, then we have that Ω̃ is a new minimizer with the first k eigenvalues strictly smaller than those in Ω. Since F is strictly increasing, we are in contradiction with the optimality of Ω. 2.5 Subsolutions for the torsion energy In order to explain the main lines of the proof of Lemma 2.9, we recall the notion of shape subsolutions introduced in [206], which allows us to replace the study of a gen- eral spectral functional with the study of the torsion energy. Roughly speaking, if a shape is optimal for a general spectral functional, it may satisfy some sub-optimality conditions for the torsion energy. From this last condition, one could deduce inter- esting qualitative properties on the optimal shape, like information on the perimeter and outer density, which is related to the boundedness. The key argument is that the variation of an eigenvalue for an arbitrary geometric perturbation of the domain can be controlled by the variation of the torsion energy, for the same perturbation (see inequality (2.8)).
- 38. 24 | Dorin Bucur Definition 2.11. We say that a quasi-open set Ω ⊂ Rd is a local shape subsolution for the torsion energy if there exists η, δ 0 such that for all quasi-open sets A ⊆ Ω with the property that ´ (wΩ − wA)dx δ we have E(Ω) + η|Ω| ≤ E(A) + η|A|. The main result proved by Bucur in [206] is the following. Theorem 2.12. Assume Ω is a local shape subsolution for the torsion energy. Then Ω is bounded and has finite perimeter. In the result above, the diameter is understood as the maximal length of the orthog- onal projection of Ω on lines. Both the diameter and the perimeter depend only on |Ω|, η, δ, d. Proof. (of Theorem 2.12) The proof of the boundedness is a consequence of the follow- ing Lemma, inspired from the seminal paper of Alt and Caffarelli [25]. Lemma 2.13. Assume Ω is a local shape subsolution for the torsion energy. There exists r0 0, C0 0 such that for every x0 ∈ Rd and r ∈ (0, r0) if sup x∈Br(x0) wΩ(x) ≤ C0r then wΩ = 0 on B 1 2 r(x0). (2.20) The proof of this lemma is classical. We refer the reader to [25], or to [206], for the specific situation of the torsion function. In order to gather information on the boundedness of a shape subsolution, one observes that for every θ 0, there exists δ0 0 depending only on N, θ such that if wΩ(x0) ≥ θ for some x0 ∈ Rd , then ˆ Bδ(x0) wΩdx ≥ wΩ(x0)ωd 2 δd , ∀δ ∈ (0, δ0). Indeed, for every x0 ∈ Rd the function x 7→ wΩ(x) + |x−x0|2 2d is subharmonic in Rd . Consequently, for every δ 0 θ ≤ wΩ(x0) ≤ 1 |Bδ| ˆ Bδ(x0) (w(x) + |x − x0|2 2d ) dx = 1 |Bδ| ˆ Bδ(x0) wdx + δ2 2(d + 2) . For δ0 sufficiently small, we have ∀ 0 δ ≤ δ0 ˆ Bδ(x0) wΩdx ≥ wΩ(x0)ωd 2 δd . As a consequence, if Ω is unbounded (or has a large diameter, in the sense that the Hausdorff measure of the projection of the set Ω on one line is large) we get that the measure is larger than any constant (depending on the length of the diameter).
- 39. 2 Existence results | 25 In order to prove that Ω has finite perimeter, we consider for every ε 0, the test function wε = (wΩ − ε)+ , which is the torsion function on the set {wΩ ε}. We get 1 2 ˆ |∇wΩ|2 dx − ˆ wΩdx + η|Ω| ≤ 1 2 ˆ |∇wε|2 dx − ˆ wεdx + η|{uε 0}|. Consequently 1 2 ˆ 0≤wΩ≤ε |∇wΩ|2 dx + η|{0 ≤ wΩ ≤ ε}| ≤ ≤ ˆ wΩ − wεdx = ˆ 0≤wΩ≤ε wΩ + ε|{wΩ ε}| ≤ ε|Ω|. By Cauchy-Schwarz ˆ 0≤wΩ≤ε |∇wΩ|dx 2 ≤ ˆ 0≤wΩ≤ε |∇wΩ|2 dx|{0 ≤ wΩ ≤ ε}| ≤ 2ε2 /η|Ω|2 , so that ˆ 0≤wΩ≤ε |∇wΩ|dx ≤ ε r 2 η |Ω|. Using the co-area formula and the average theorem, we find εn 0, εn → 0 such that Hd−1 (∂* {wΩ εn}) ≤ r 2 η |Ω|, where ∂* Ω denotes the measure theoretic boundary. Passing to the limit, we get Hd−1 (∂* Ω) ≤ r 2 η |Ω|. This last inequality implies that Ω has a finite perimeter in the geometric measure theoretical sense. Sketch of the proof of Lemma 2.9. Given Ω ⊆ Rd , a quasi-open set of volume c, we find first a solution of the following problem min{E(A) + η|A| : A ⊆ Ω}, for a suitably chosen value η 0, which will be fixed later. For every η 0, this problem has a solution. The existence can be proved by the direct method of the calculus of variations, as a consequence of the compact embed- ding H1 0(Ω) in L2 (Ω). In fact any weak gamma limit of a minimizing sequence (An)n is a solution. Let us denote Ωη a solution. We define the set Ω̃ = |Ω| |Ωη| 1 d Ωη.
- 40. 26 | Dorin Bucur Since Ωη is a subsolution for the torsion energy, the diameter and perimeter of Ω̃ are controlled only by η, c and d. It is easy to notice that if η is small enough (the precise value will be fixed at the end), then the first k eigenvalues of Ω̃ are not larger than the corresponding eigenval- ues on Ω. This is essentially a consequence of inequality (2.8) λk(Ωη) − λk(Ω) ≤ 4k2 e1/4π λk(Ωη)λk(Ω)(d+2)/2 [E(Ω) − E(Ωη)]. (2.21) Indeed, if η is small enough, one can prove (see (2.10)-(2.11)) that λk(Ωη) ≤ 2λk(Ω) and get λk(Ωη) − λk(Ω) ≤ Cη,λk(Ω),c,d(|Ω| 2 d − |Ωη| 2 d ). (2.22) The constant Cη,λk(Ω),c,d is smaller when η and λk(Ω) are smaller. The dependence of Cη,λk(Ω),c,d on all parameters, including η is explicit. As a consequence, we get λk(Ωη) + Cη,λk(Ω),c,d|Ωη| 2 d ≤ λk(Ω) + Cη,λk(Ω),c,d|Ω| 2 d , (2.23) which leads for any value Cη,λk(Ω),c,d ≤ λk(Ω) c 2 d to the inequality λk(Ωη)|Ωη| 2 d ≤ λk(Ω)|Ω| 2 d . (2.24) This implies that λk(Ω̃) ≤ λk(Ω). Clearly, inequalities λi(Ω̃) ≤ λi(Ω) also hold for i = 1, . . . , k − 1. Moreover, the set Ωη being a subsolution for the torsion energy, it is bounded and has finite perimeter, controlled only by Ω, η and d. This holds as well for the set Ω̃, with rescaling factors coming from the ratio |Ω| |Ωη| . Provided that the constant η is chosen small enough such that this ratio is not larger than 2 and that Cη,λk(Ω),c,d ≤ λk(Ω) c 2 d , we conclude the proof. Further remarks The perimeter constraint. A natural question is to ask if Theorem 2.10 could hold under a further constraint Per(Ω) ≤ c2. Of course, this question becomes interesting, as soon as the constant c2 is smaller than the perimeter of the optimal set in Theorem 2.10. In order to deal with these kind of questions, in [221] a second surgery result is proved, with the purpose of having a finer control of the perimeter. This result asserts, roughly speaking, that one can also decrease the perimeter of a set if its diameter is large in Lemma 2.9. For this purpose, the surgery procedure is performed in a different way.
- 41. 2 Existence results | 27 Lemma 2.14. (surgery of the perimeter) For every K, P, c 0, there exist D 0 depending only on K, P, c and the dimension d, such that for every quasi-open set Ω ⊂ Rd with |Ω| = c, Per(Ω) ≤ P, there exists a quasi-open set Ω̃ of the same measure, with diam (Ω̃) ≤ D, Per(Ω̃) ≤ Per(Ω) such that if for some k ∈ N it holds λk(Ω) ≤ K, then λi(Ω̃) ≤ λi(Ω) for all 1 ≤ i ≤ k. A consequence of this lemma concerns the following spectral optimization problem min{F(λ1(Ω), . . . , λk(Ω)) : Ω ⊆ Rd , |Ω| = c1, Per(Ω) ≤ c2}. (2.25) Theorem 2.15. Provided that F : Rk → R is non-decreasing in each variable and lower semicontinuous, for every c1, c2 0 such that c2 ≥ Hd−1 (∂B1) |B1| d−1 d c d−1 d 1 , problem (2.25) has a solution in the class of measurable sets. We refer to [330] for details on shape optimization problems on measurable sets. Roughly speaking, this means that the minimum is attained on a quasi-open set Ω, for which there exists a measurable set A such that Ω ⊆ A, |A| = c1 and Per(A) ≤ c2 (see [221] for details). Optimization in specific classes of sets. An interesting task is to search for the ex- tremal sets of spectral functionals in some specific classes of sets, e.g. the class of convex subsets of Rd (satisfying, or not, a constraint on measure, perimeter or diam- eter), the class of simply connected sets open sets of R2 , the class of N-gones of R2 , etc. As a general fact, one can notice that the existence question has a much more direct answer, as soon as those geometric or topological constraints are imposed. For instance, Theorems 2.1, 2.10 can be rephrased in the class of open convex sets (in any dimension of the space) or in the class of open sets in R2 whose complement have at most l connected components (l is a fixed natural number) (see [207, Sections 4.6, 4.7 and Chapters 5, 6]). In the family of convex sets, very interesting phenomena may occur leading to optimal sets which are locally of polygonal type. We refer the reader to [622, 623] and Section 3.5 of Chapter 3. Other boundary conditions, higher order operators. In this chapter we discussed the existence questions only for functionals depending on the spectrum of the Laplace operator with Dirichlet boundary conditions. A good question is whether or not similar results hold for the Laplace operator with other boundary conditions. Working with different boundary conditions requires us to completely change the functional framework. For instance, when optimizing spectral functionals associated with the Neumann Laplacian, similar techniques as in the proof of Theorem 2.1 can hardly be used. In fact, the functional space one has to use for the Neumann Lapla- cian is H1 (Ω). For different sets Ω, the spaces H1 (Ω) are not naturally embedded in a good functional space, unless (uniform) geometric requirements are satisfied by
- 42. 28 | Dorin Bucur the different sets. The existence question for general functionals depending on the spectrum of the Neumann Laplacian is completely open (see [207, Chapter 7]). For Robin boundary conditions (or for the Steklov problem) one could use the theory of special functions of bounded variations in order to handle existence, at least in some specific situations. The regularity of the boundaries of the optimal sets, relies here on the theory of free discontinuity problems. We refer the reader to the discussion around Theorem 4.24 in Chapter 4 and to [216, 217] for an introduction to the topic. For the bi-Laplace operator with Dirichlet boundary conditions, a similar result to Theorem 2.1 holds true in a bounded design region, while for other boundary condi- tions or D = Rd , the question is open. Asymptotic behavior for large k. An interesting question is to understand the be- havior of a sequence of solutions of problem (2.5) when k goes to +∞. Only partial answers are known: in two dimensions of the space, if the measure constraint is re- placed by a perimeter constraint, then any sequence of optimal domains converges to the disc, as it was recently proved by Bucur and Freitas. The question is to understand if a similar result continues to be true for the measure constraint, in any dimension of the space. A key problem is to prove that all the optimal sets for problem (2.5) are uniformly bounded, independently on k. Even partial results, asserting that subse- quences of solutions have a geometric limit, would be of interest.
- 43. Jimmy Lamboley and Michel Pierre 3 Regularity of optimal spectral domains 3.1 Introduction The main goal of this chapter is to review known results and open problems about the regularity of optimal shapes for the minimization problems min λk(Ω), Ω ⊂ D, |Ω| = a , (3.1) where D is a given open subset of Rd , | · | denotes the Lebesgue measure, a ∈ (0, |D|), k ∈ N* and λk(Ω) is the k-th eigenvalue of the Laplace operator on Ω with ho- mogeneous Dirichlet boundary conditions. We will also consider the regularity ques- tion for penalized versions of (3.1) and discuss the possibility of singularities appear- ing for optimal shapes, either for (3.1) or for related problems involving convexity con- straints. We refer to Chapter 2 for all of the necessary definitions and for the question of existence of optimal shapes. In particular, if D is bounded or if D = Rd , then Problem (3.1) has a solution (say Ω* ) in the family of quasi-open subsets of Rd (as explained in Chapter 2, the eigenvalues λk(Ω) may be well-defined for all quasi-open sets Ω with finite measure as well as the space H1 0(Ω)). We analyze the question of the regularity of this optimal shape Ω* . This turns out to be a difficult and wide open question. It is even difficult to decide whether Ω* is open or not and this is not completely understood yet. Is Ω* always open? Is at least one of the optimal Ω* open? What is the regu- larity of the optimal k-th eigenfunctions uΩ* ? As recalled in Chapter 1 and Chapter 2, if D = Rd or more generally if D is ’large enough’, then we know that: – Ω* is a ball if k = 1, – Ω* is the union of two disjoint identical balls if k = 2, Jimmy Lamboley: CEREMADE, Université Paris-Dauphine, CNRS, PSL Research University, 75775 Paris, France, E-mail: lamboley@ceremade.dauphine.fr Michel Pierre: IRMAR, ENS Rennes, CNRS, UBL, av Robert Schumann, 35170 - Bruz, France, E-mail: michel.pierre@ens-rennes.fr © 2017 Jimmy Lamboley and Michel Pierre This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
- 44. 30 | Jimmy Lamboley and Michel Pierre with uniqueness in both cases up to translations (and sets of zero-capacity). Here, D ’large enough’ means that, when k = 1, it can contain a ball of volume a, and when k = 2, it can contain two disjoint identical balls whose total volume is a. Thus full regularity holds for the optimal shape in these two cases. The question remains however open for ’large’ D with k ≥ 3 and for any k with ’small’ D. Then, the regularity analysis of the optimal shapes in (3.1) is very similar to the analysis of the optimal shapes for the Dirichlet energy, namely min Gf (Ω), Ω ⊂ D, |Ω| = a , (3.2) where f ∈ L∞ (D) is given and Gf (Ω) = ˆ Ω 1 2 |∇uΩ|2 − f uΩ , uΩ ∈ H1 0(Ω), −∆uΩ = f in Ω. (3.3) (The solution uΩ of this Dirichlet problem is classically defined when Ω is an open set with finite measure. As explained in Chapter 2, this definition may be extended to the case when Ω is only a quasi-open set with finite measure.) Actually, for these two problems (3.1) and (3.2), the analysis of the regularity fol- lows the same main steps and offers the following main features. They will provide the content of Sections 3.2 and 3.3. 1. The situation is easier when the state function is nonnegative ! For the Dirich- let energy case (3.2), for instance in dimension two, full regularity of the boundary holds for positive data f, inside D (see [187] and Paragraph 3.2.3.1 below). On the other hand, even in dimension two, it is easily seen that singularities do necessar- ily occur at each point of the boundary of the optimal set Ω* in the neighborhood of which the state function uΩ* (as defined in (3.3)) changes sign. The change of sign of uΩ* does imply that its gradient has to be discontinuous and, therefore, that the boundary cannot be regular near these points. For instance, cusps will then generally occur in dimension two (see e.g. [509]). For the eigenvalue problem (3.1), state functions are the k-th eigenfunctions on Ω* of the Laplace-Dirichlet operator. Thus the situation (and the analysis) will be quite different if k = 1 where the first eigenfunction is nonnegative and if k ≥ 2 where the eigenfunction changes sign. This partly explains why we devote the specific Section 3.2 to Problem (3.1) with k = 1. One more specific feature is that the problem is then equivalent to a minimization problem where the variables are functions rather than domains and we are led to a free boundary formulation (see Paragraph 3.2.1) where one has to understand the regularity of the boundary of [uΩ* 0]. One can essentially obtain as good of regularity results as one did with the Dirichlet energy case and nonnegative data f, see [189]. Here we strongly rely on the seminal paper [25] by Alt-Caffarelli about regularity of free boundaries. On the other hand, the case k ≥ 2 is far from being so well understood and we will try to describe what current state of the art is (see Section 3.3).
- 45. 3 Regularity of optimal spectral domains | 31 2. A first step: regularity of the state function. For the Dirichlet energy case, the analysis starts by studying the regularity of uΩ* as defined in (3.3). It is proved (see [188]) that uΩ* is locally Lipschitz continuous on D, for any optimal shape Ω* and no matter the sign of uΩ* . This Lipschitz continuity is the optimal regularity we can expect for uΩ* , as it vanishes on D Ω* , and is expected to have a non vanish- ing gradient on ∂Ω* from inside Ω* . As expected, the proof in the case where uΩ* changes sign is much more involved and requires for instance the Alt-Caffarelli- Friedman Monotonicity Lemma (proved in [26], [245], see Lemma 3.36 below). For the optimal eigenvalue problem (3.1) with k = 1, it can be proved as well that the corresponding eigenfunction on Ω* is locally Lipschitz continuous on D (see Theorem 3.16). For k ≥ 2 and D = Rd , it has been proved in [222] that one of the k-th eigenfunctions is Lipschitz continuous (see Theorem 3.35) (note that the optimal eigenvalue is generally expected to be of multiplicity higher than once). However, in the case where D is bounded and k ≥ 2, the problem is still not understood. The main difference is that, when D = Rd , Problem (3.1) is equivalent to the penalized version min n λk(Ω) + µ|Ω|, Ω ⊂ Rd o , (3.4) for some convenient µ ∈ (0, ∞) (see Proposition 3.33). More regularity information may then be derived on optimal state functions for penalized versions (see below). 3. Penalized versions. In order to obtain information on the regularity of Ω* or uΩ* , we consider admissible perturbations of Ω* and use their minimization proper- ties. Obviously, there is more freedom in choosing perturbations on the penalized version (3.4) where the volume constraint |Ω| = a is relaxed, rather than on the constrained initial version (3.1). The analysis of (3.1) when k = 1 starts by showing that (3.1) is equivalent to the penalized version min λ1(Ω) + µ[|Ω| − a]+ , Ω ⊂ D , (3.5) for µ large enough (see Proposition 3.7). Analysis of the regularity may then be more easily made on the optimal shapes of (3.5). In Paragraph 3.2.3.2, we make an heuristic analysis of this “exact penalty” property for general optimization prob- lems where not only the penalized version converges to the constrained problem as the penalization coefficient µ → ∞, but more precisely that the two prob- lems are equivalent for µ large enough. Optimal such factors µ play the role of La- grange multipliers. This approach is used again in a local way in Paragraph 3.2.3.3, to prove that the ’pseudo’-Lagrange multiplier does not vanish (see Proposition 3.24). It is also used in Chapter 7 of this book to study the regularity of optimal shapes for similar functionals (see Step 5 in the proof of Theorem 7.13). 4. How to obtain the regularity of the boundary of Ω* ? Knowing that the state function is Lipschitz continuous is a first main step in the study of the regularity of the boundary of the optimal set, but obviously not sufficient.
- 46. 32 | Jimmy Lamboley and Michel Pierre For example when k = 1, this boundary can be seen as the boundary of the set [uΩ* 0]. If we were in a regular situation (say if u were C1 on Ω*), then knowing that the gradient of uΩ* does not vanish at the boundary would imply regularity of this boundary by the implicit function theorem. Indeed, the next main step is (heuristically) to prove that the gradient of the state function does not degenerate at the boundary. This is what is done and then used in Paragraph 3.2.3.3 for the optimal sets of (3.1) when k = 1. Full regularity of the boundary is proved in dimension two and regularity of the reduced boundary is proved in any dimension (see Theorem 3.20). Here we strongly rely on the seminal paper [25] by Alt-Caffarelli as explained in details in Section 3.2.3.1. Note that it is also used in Chapter 7 of this book as mentioned at the end of Point 3 above. Noth- ing like this is known when k ≥ 2. It is already a substantial piece of information to sometimes know that Ω* is an open set ! (see Section 3.3). In Section 3.4, we partially analyze the regularity of Ω* solution of (3.1) up to the boundary of the box D, when k = 1. We notice in particular that it is natural to expect the contact to be tangential (although this is not proved anywhere as far as we know), but we cannot expect in general that the contact will be very smooth; we prove that when D is a strip (too narrow to contain a disc of volume a), the optimal shape is C1,1/2 and not C1,1/2+ε with ε 0. In order to show that this behavior is not exceptional and is not only due to the presence of a box constraint, we show that a similar property is valid for solutions to the problem min λ2(Ω), Ω open and convex, |Ω| = a . This last problem enters the general framework of convexity constraints, which is quite challenging from the point of view of calculus of variations. We conclude this chapter with Section 3.5 where we discuss some problems in this framework. They are of the form min J(Ω), Ω open and convex , where J involves λ1, and possibly other geometrical quantities (such as the volume |Ω| or the perimeter P(Ω)), and which lead to singular optimal shapes, such as polygons (in dimension 2). Thanks to the convexity constraint, we are allowed to consider the question of maximizing the perimeter and/or the first Dirichlet eigenvalue, and in this direction we discuss a few recent results about the reverse Faber-Krahn inequality. Remark 3.1. The question of regularity could also be considered for the following optimization problems: min λk(Ω), Ω ⊂ D, P(Ω) = p , min P(Ω) + λk(Ω), Ω ⊂ D, |Ω| = a where P denotes the perimeter (in the sense of geometric measure theory), and D is either a bounded smooth box, or Rd . In these cases, it has been shown in [329, 330]
- 47. 3 Regularity of optimal spectral domains | 33 that the regularity of optimal shapes is driven by the presence of the perimeter term. More precisely it can be shown that they exist (which is not trivial if D = Rd ) and that they are quasi-minimizers of the perimeter, and therefore smooth outside a singular set of dimension less than d − 8. 3.2 Minimization for λ1 In this section, we focus on the regularity of the optimal shapes of the following prob- lem: min λ1(Ω), Ω ⊂ D, Ω quasi − open, |Ω| = a , (3.6) where D is an open set in Rd , a ∈ (0, |D|) and k ∈ N* . Thanks to the Faber-Krahn inequality, it is well-known that, if D contains a ball of volume a, then this ball is a solution of the problem, and is moreover unique, up to translations (and to sets of zero-capacity). Therefore, the results of this section are relevant only if such a ball does not exist. 3.2.1 Free boundary formulation We first give an equivalent version of problem (3.6) as a free boundary problem, namely an optimization problem in H1 0(D) where domains are level sets of functions. Notation. For w ∈ H1 0(D), we will denote Ωw = {x ∈ D; w(x)= 6 0}. Recall that for a bounded quasi-open subset Ω of D (see Chapter 2) λ1(Ω) = min ˆ Ω |∇v|2 ; v ∈ H1 0(Ω), ˆ Ω v2 = 1 . (3.7) Definition 3.2. In this section, we denote by uΩ any nonnegative minimizer in (3.7), i.e. such that uΩ ∈ H1 0(Ω), ˆ Ω |∇uΩ|2 = λ1(Ω), ˆ Ω u2 Ω = 1. Remark 3.3. Choosing in (3.7) v = v(t) := (uΩ + tφ)/kuΩ + tφkL2(Ω) with φ ∈ H1 0(Ω), and using that the derivative at t = 0 of t 7→ ´ Ω |∇v(t)|2 vanishes leads to ∀ φ ∈ H1 0(Ω), ˆ Ω ∇uΩ∇φ = λ1(Ω) ˆ Ω uΩφ. (3.8) If Ω is an open set, (3.8) means exactly that −∆uΩ = λ1uΩ in the sense of distributions in Ω. Note that if uΩ is a minimizer in (3.7), so is |uΩ|. Therefore, with no loss of gen- erality, we can assume that uΩ ≥ 0 and we will always make this assumption in this
- 48. 34 | Jimmy Lamboley and Michel Pierre section on the minimization of λ1(Ω). If Ω is a connected open set, then uΩ 0 on Ω. This is a consequence of the maximum principle applied to −∆uΩ = λ1(Ω)uΩ ≥ 0 on Ω. This extends (quasi-everywhere) to the case when Ω is a quasi-connected quasi-open set, but the proof requires a little more computation. Since Ω 7→ λ1(Ω) is nonincreasing with respect to inclusion, any solution of (3.6) is also solution of min λ1(Ω), Ω ⊂ D, Ω quasi − open, |Ω| ≤ a . (3.9) The converse is true in most situations, in particular if D is connected, see Remark 3.6, Corollary 3.18 and the discussion in Section 3.2.4.1. Note that it may happen that if D is not connected, then a solution to (3.9) does not satisfy |Ω| = a. We will first consider Problem (3.9) and this will nevertheless provide a complete understanding of (3.6). We start by proving that (3.9) is equivalent to a free boundary problem. Proposition 3.4. 1. Let Ω* be a quasi-open solution of the minimization problem (3.9) and let u = uΩ* . Then ˆ D |∇u|2 = min ˆ D |∇v|2 ; v ∈ H1 0(D); ˆ D v2 = 1, |Ωv| ≤ a . (3.10) 2. Let u be solution of the minimization problem (3.10). Then Ωu is solution of (3.9). Proof. For the first point, we choose v ∈ H1 0(D) with |Ωv| ≤ a and we apply (3.9) to Ω = Ωv. This gives ´ D |∇u|2 = λ1(Ω* ) ≤ λ1(Ωv) and we use the property (3.7) for λ1(Ωv) so that ˆ D |∇u|2 ≤ min ˆ D |∇v|2 ; v ∈ H1 0(D), ˆ D v2 = 1, |Ωv| ≤ a . Equality holds since u ∈ H1 0(Ω* ) ⊂ H1 0(D), and |Ωu| = |Ω* | ≤ a. For the second point, let u be a solution of (3.10). Then, |Ωu| ≤ a, ´ D u2 = 1. Let Ω ⊂ D quasi-open with |Ω| ≤ a and let uΩ as in Definition 3.2. Then λ1(Ωu) ≤ ˆ D |∇u|2 ≤ ˆ D |∇uΩ|2 = λ1(Ω). Remark 3.5. We will now work with the functional problem (3.10) rather than (3.9). Note that if D is bounded (or with finite measure), then existence of the minimum u follows easily from the compactness of H1 0(D) into L2 (D) applied to a minimizing se- quence (that we may assume to be weakly convergent in H1 0(D) and strongly in L2 (D)). Remark 3.6. Two different situations may occur. If D is connected and Ω* solves (3.9), then a* := |[uΩ* 0]| = a and Ω* = [uΩ* 0]. If D is not connected, it may happen
- 49. 3 Regularity of optimal spectral domains | 35 that a* a and therefore uΩ* 0 on some of the connected components of D and identically zero on the others. Indeed, if a* a, then for all balls B ⊂ D with measure less than a − a* and all φ ∈ H1 0(B), we may choose v = v(t) = (u + tφ)/ku + tφkL2(D) with u := uΩ* ≥ 0 in (3.10). Writing that the derivative at t = 0 of t 7→ ´ D |∇v(t)|2 vanishes gives ˆ D ∇u∇φ = λa ˆ D u φ with λa := ˆ D |∇u|2 , and this implies : −∆u = λau in D. The strict maximum principle implies that, in each connected component of D, either u 0 or u ≡ 0. If D is connected, we get a contra- diction since a |D|. Therefore necessarily a* = a if D is connected. We refer to Corollary 3.18 and Proposition 3.29 for a complete description of the regularity when D is not connected. 3.2.2 Existence and Lipschitz regularity of the state function 3.2.2.1 Equivalence with a penalized version We will first prove that (3.10) is equivalent to a penalized version. Proposition 3.7. Assume |D| +∞. Let u be a solution of (3.10) and λa := ´ D |∇u|2 . Then, there exists µ 0 such that ˆ D |∇u|2 ≤ ˆ D |∇v|2 + λa 1 − ˆ D v2 + + µ [|Ωv| − a] + , ∀v ∈ H1 0(D). (3.11) Remark 3.8. Given a quasi-open set Ω ⊂ D, and choosing v = uΩ in (3.11), we obtain the penalized ’domain’ version of (3.9), where Ω* is solution of (3.9) λ1(Ω* ) ≤ λ1(Ω) + µ[|Ω] − a]+ , ∀ Ω ⊂ D, Ω quasi − open. (3.12) Proof of Proposition 3.7.. Note first that, by definition of u and of λa, for all v ∈ H1 0(D) with |Ωv| ≤ a, we have ´ D |∇v|2 − λa ´ D v2 ≥ 0, or ˆ D |∇u|2 ≤ ˆ D |∇v|2 + λa 1 − ˆ D v2 . (3.13) Let us now denote by Jµ(v) the right-hand side of (3.11) and let uµ be a minimizer of Jµ(v) for v ∈ H1 0(D) (its existence follows by compactness of H1 0(D) into L2 (D), see also Remark 3.5). Up to replacing uµ by |uµ|, we may assume uµ ≥ 0. Using that Jµ(uµ) ≤ Jµ(uµ/kuµk2), we also deduce that kuµk2 2 = ´ D u2 µ ≤ 1. For the conclusion of the proposition, it is sufficient to prove |Ωuµ | ≤ a since then Jµ(uµ) ≤ Jµ(u) = ˆ D |∇u|2 ≤ Jµ(uµ),
- 50. 36 | Jimmy Lamboley and Michel Pierre where this last inequality comes from (3.13). In order to obtain a contradiction, assume that |Ωuµ | a and introduce ut := (uµ − t)+ . Then Jµ(uµ) ≤ Jµ(ut ). This implies, using |Ωut | a for t small, that ˆ [0uµ t] |∇uµ|2 + µ [0 uµ t] ≤ λa ˆ [0uµ t] u2 µ + 2tλa ˆ D uµ. Using the coarea formula (see e.g. [370], [432]), this may be rewritten for t ≤ t0 ≤ p µ/λa as ˆ t 0 ds ˆ [uµ=s] |∇uµ| + µ − λas2 |∇uµ| dHd−1 ≤ 2tλa ˆ D uµ ≤ 2tλa|Ωuµ |1/2 . But the function x ∈ (0, ∞) 7→ x + (µ − λas2 )x−1 ∈ [0, ∞) is bounded from below by 2 p µ − λas2 and also by 2 q µ − λat2 0 as soon as s2 ≤ t2 ≤ t2 0 ≤ µ/λa. Therefore, it follows that ∀ t ∈ [0, t0), 2 q µ − λat2 0 ˆ t 0 ˆ [uµ=s] dHd−1 ≤ 2tλa|Ωuµ |1/2 . (3.14) We now use the isoperimetric inequality: ´ [uµ=s] dHd−1 ≥ C(d) [uµ s] d−1 d . We divide the inequality by t and we let t → 0, then t0 → 0, to deduce 2 √ µ C(d)|Ωuµ | d−1 d ≤ 2λa|Ωµ|1/2 , and finally 2 √ µ C(d) a d−2 2d ≤ 2λa. Thus, if d ≥ 2, |Ωuµ | a is impossible if µ µ* := λ2 aC(d)−2 a(2−d)/d . Therefore the conclusion of Proposition 3.7 holds for any µ µ* . If d = 1, we have √ µC(1) ≤ λa|Ωµ|1/2 . On the other hand, by definition of uµ we also have |Ωuµ | ≤ a + λ1(Ω1)/µ for some fixed Ω1 ⊂ D with |Ω1| = a. We deduce an upper bound for µ as well. Remark 3.9. With respect to the heuristic remarks made in Paragraph 3.2.3.2, it is interesting to notice that our problem here is not in a ’differentiable setting’. However, we do perform some kind of differentiation in the direction of the perturbations t 7→ (uµ − t)+ . This provides the upper bound µ* on µ which plays the role of a Lagrange multiplier. This remark is a little more detailed in Paragraph 3.2.3.2. Note that µ* does not depend on |D|. The assumption |D| ∞ was used only to prove existence of the minimizer uµ. Remark 3.10 (Sub- and super-solutions). Note that to prove Proposition 3.7, we only use perturbations of the optimal domain Ωu from inside. This means that the same result is valid for shape subsolutions where (3.10) is assumed only for functions v for which Ωv ⊂ Ωu. Next, we will prove Lipschitz continuity of the functions u solutions of the penal- ized problem (3.11). Interestingly, Lipschitz continuity will hold for super-solutions of (3.11) which are defined when the inequality (3.11) is valid only for perturbations from outside, i.e. such that Ωu ⊂ Ωv.
- 51. 3 Regularity of optimal spectral domains | 37 3.2.2.2 A general sufficient condition for Lipschitz regularity We now state a general result to prove Lipschitz regularity of functions independently of shape optimization. It applies to signed functions as well and will be used again in the minimization of the k-th eigenvalue. Proposition 3.11. Let U ∈ H1 0(D), bounded and continuous on D and let ω := {x ∈ D; U(x)= 6 0}. Assume ∆U is a measure such that ∆U = g on ω with g ∈ L∞ (ω) and |∆|U|| (B(x0, r)) ≤ Crd−1 (3.15) for all x0 ∈ D with B(x0, 2r) ⊂ D, r ≤ 1 and U(x0) = 0. Then U is locally Lipschitz continuous on D. If moreover D = Rd , then U is globally Lipschitz continuous. Remark 3.12. Note that if U is locally Lipschitz continuous on D with ∆U ≥ 0, then for a test function φ with φ ∈ C∞ 0 (B(x0, 2r)), B(x0, 2r) ⊂ D, 0 ≤ φ ≤ 1, φ ≡ 1 on B(x0, r), k∇φkL∞(B) ≤ C/r, (3.16) we have ∆U(B(x0, r)) ≤ ˆ D φd(∆U) = − ˆ D ∇φ∇U ≤ k∇UkL∞ |Ωφ|k∇φkL∞ ≤ Ck∇UkL∞ rd−1 . This indicates that the estimate (3.15) is essentially a necessary condition for the Lips- chitz continuity of U. This theorem states that the converse holds in some cases which are relevant for our analysis as it will appear in the next paragraph. Remark 3.13. In the proof of Proposition 3.11, as in [188], we will use the following identity which is useful to estimate the variation of functions: ∂B(x0,r) U(x)dσ(x) − U(x0) = C(d) ˆ r 0 s1−d ˆ B(x0,s) d(∆U) ds. (3.17) This is easily proved for regular functions U by integration in s of d ds ∂B(0,1) U(x0 + sξ)dσ(ξ) = ∂B(0,1) ∇U(x0 + sξ) · ξ = C(d)s1−d ˆ B(x0,s) ∆U, which implies that for a.e. 0 r1 r2, ∂B(x0,r2) U(x)dσ(x) − ∂B(x0,r1) U(x)dσ(x) = C(d) ˆ r2 r1 s1−d ˆ B(x0,s) ∆U ds. It extends to functions U ∈ H1 (D) where ∆U is a measure with ´ r 0 s1−d ´ B(x0,s) d(|∆U|)ds ∞. We may then consider that U is precisely defined at x0 as: U(x0) = lim r→0+ ∂B(x0,r) U(x)dσ(x), (3.18)
- 52. 38 | Jimmy Lamboley and Michel Pierre and (3.17) holds with this precise definition of U(x0). Proof of Proposition 3.11. We want to prove that ∇U ∈ L∞ loc(D). We can first claim that ∇U = 0 a.e. on D ω. On the open set ω, we have ∆U = g ∈ L∞ (ω) so that at least U ∈ C1 (ω). Let us denote Dδ = {x ∈ D; d(x, ∂D) δ} (we start with the case D= 6 Rd ). We will bound ∇U(x0) for x0 ∈ ω∩Dδ. The meaning of the constant C will vary but always de- pend only on δ, kUkL∞(D), kgkL∞(D), d and on the constant C in the assumption (3.15). Let y0 ∈ ∂ω be such that |x0 − y0| = d(x0, ∂ω) := r0. Then r0 0 and B(x0, r0) ⊂ ω. We have U(y0) = 0 since y0 ∈ ∂ω and U is continuous. Let us introduce s0 := min{r0, 1}, B0 := B(x0, s0) and V ∈ H1 0(B0) such that ∆V = g on B0. Since g ∈ L∞ , by scaling we obtain kVkL∞(B0) ≤ Cs2 0, k∇VkL∞(B0) ≤ Cs0, C = C(kgkL∞ ). Since U − V is harmonic on B0, we also have |∇(U − V)(x0)| ≤ d s0 kU − VkL∞(B0) so that |∇U(x0)| ≤ |∇V(x0)| + ds−1 0 kU − VkL∞(B0) ≤ C h s0 + s−1 0 kUkL∞(B0) i . (3.19) If s0 ≥ δ/16, we deduce from (3.19): |∇U(x0)| ≤ C(δ, kUkL∞ , kgkL∞ ). We now assume δ ≤ 16. If s0 δ/16 i.e. r0 = s0 δ/16, since x0 ∈ Dδ, d(y0, ∂D) ≥ d(x0, ∂D) − d(x0, y0) ≥ δ − r0 ≥ 15r0 which implies B(x0, r0) ⊂ B(y0, 2r0) ⊂ B(y0, 8r0) ⊂ D. Thanks to assumption (3.15), U(y0) = 0 and to formula (3.17) applied with U replaced by |U|, we deduce ffl ∂B(y0,4r0) |U(z)|dσ(z) ≤ C r0. Finally, using the representation (U − V)(x) = ffl B(y0,4r0) U(z)Px(z)dσ(z) for all x ∈ B(y0, 2r0) where Px(·) is the Poisson kernel at x, we have kU − VkL∞(B0) ≤ kU − VkL∞(B(y0,2r0)) ≤ C ∂B(y0,4r0) |U(z)| dσ(z) ≤ C r0. This together with (3.19) (where s0 = r0) and kVkL∞(B0) ≤ Cr2 0, this implies |∇u(x0)| ≤ C. Now if D = Rd , either ω = Rd and (3.19) gives the estimate (r0 = +∞, s0 = 1), or ω= 6 Rd : then we argue just as above, replacing δ/16 by 1 in the discussion. In Proposition 3.11, the function U is assumed to be continuous on D. For our optimal eigenfunctions, this will be a consequence of the following lemma. Lemma 3.14. Let U ∈ H1 0(D) such that ∆U is a measure satisfying |∆U| B(x0, r) ≤ Crd−1 , (3.20) for all x0 ∈ D with B(x0, 2r) ⊂ D, r ≤ 1. Then U is continuous on D. Proof. Assumption (3.20) implies that ´ r 0 s1−d |∆U|(B(x0, s)) ∞ so that (3.18) and (3.17) hold. Let x0, y0 ∈ D and r 0 small enough so that B(x0, 2r) ⊂ D, B(y0, 2r) ⊂ D.
- 53. 3 Regularity of optimal spectral domains | 39 We deduce, using (3.20) again and the representation (3.18): |U(x0) − U(y0)| ≤ ffl ∂B(x0,r) U − ffl ∂B(y0,r) U + C r ≤ ffl ∂B(0,r) |U(x0 + ξ) − U(y0 + ξ)|dσ(ξ) + C r. But by continuity of the trace operator from H1 (B(0, r)) into L1 (B(0, r)), this implies |U(x0) − U(y0)| ≤ C(r)kU(x0 + .) − U(y0 + .)kH1(B(0,r)) + C r. Thus lim sup y0→x0 |U(x0) − U(y0)| ≤ C r. Since this is valid for all r sufficiently small, continuity of U at x0 follows and therefore continuity on D as well. Remark 3.15. Looking at the proof, we easily see that the assumptions could be weak- ened in Lemma 3.14: U ∈ W1,1 0 (D) would be sufficient and rd−1 could be replaced in (3.20) by rd−2 ε(r) with ε(r)/r integrable on (0, 1). 3.2.2.3 Lipschitz continuity of the optimal eigenfunction Theorem 3.16. Let u be a solution of (3.10). Then u is locally Lipschitz continuous on D. Proof. Up to replacing u by |u|, we may assume that u ≥ 0. We will first show that U = u satisfies the assumptions of Lemma 3.14. It will follow that u is continuous on D. Therefore, we will have −∆u = λau on the open set ω = [u 0] (see Remark 3.3). Then we will prove (see also Remark 3.17 below) that − ∆u ≤ λau in D. (3.21) This will imply that ∆u is a measure and also, by an easy bootstrap that u ∈ L∞ (D). Thus the assumptions of Proposition 3.11 will be satisfied and local Lipschitz continu- ity on D will follow. By Proposition 3.7, u is also solution of Problem (3.11). We apply this inequality with v = u + tφ, t 0, φ ∈ H1 0(D). Then 0 ≤ ˆ D 2∇u∇φ + t|∇φ|2 + λa h −2uφ − tφ2 i+ + µ t |Ωu+tφ| − a + . (3.22) Choosing first φ = −pn(u)ψ where ψ ∈ C∞ 0 (D), ψ ≥ 0, and pn(r) = min{r+ /n, 1}, we obtain with qn(r) = ´ r 0 pn(s)ds and after letting t → 0 (note that |Ωu+tφ| = |Ωu| ≤ a) 0 ≤ ˆ D −2p′n(u)|∇u|2 ψ − 2∇qn(u)∇ψ + 2λaupn(u)ψ.
- 54. 40 | Jimmy Lamboley and Michel Pierre Note that upn(u) → u+ = u, qn(u) → u+ = u in a nondecreasing way as n increases to +∞. Using p′n(u)|∇u|2 ≥ 0, we obtain at the limit that ∆u + λau ≥ 0 in the sense of distributions in D, whence (3.21). Choosing φ ∈ C∞ 0 (D)+ in (3.22) leads to −2 ´ D ∇u∇φ ≤ ´ D t|∇φ|2 + µ t |Ωφ| or 2h∆u + λau, φi ≤ ˆ D 2λauφ + t|∇φ|2 + µ t |Ωφ|. (3.23) Minimizing over t ∈ (0, ∞) gives h∆u + λa, φi ≤ ˆ D λauφ + k∇φkL2 [µ|Ωφ|] 1/2 . (3.24) Let now x0 ∈ D such that B(x0, 2r) ⊂ D and let φ ∈ C∞ 0 (B(x0, 2r))+ as in (3.16). Using also u ∈ L∞ , we deduce that |∆u| B(x0, r) ≤ (∆u + λau) (B(x0, r) + λa ˆ B(x0,r) u ≤ Crd−1 , whence the estimate (3.15). Remark 3.17. Here, we use the positivity of u. Actually, u is an eigenfunction for the eigenvalue λa on Ωu. Since Ωu is open, we know that ∆u + λau = 0 on Ωu (see Remark 3.3). Since u ≥ 0, one can prove that ∆u + λau ≥ 0 on D. To prove this, use the test functions φ = −pn(u)ψ which satisfy Ωφ ⊂ Ωu and therefore belong to H1 0(Ωu). Thus applying (3.8) in Remark 3.3 with this φ is sufficient (and we finish as above). This positivity of the measure ∆u + λau allows to directly estimate the mass of |∆u| on balls only with the information (3.24). This will not be the case when dealing with k-th eigenfunctions when k ≥ 2 (see the remarks and comments on the use of the Monotonicity Lemma 3.36). Let us now state a corollary of Proposition 3.16 for the initial actual shape optimization problem (3.6). Corollary 3.18. Assume D is open and with finite measure. Then there exists an open set Ω* which is solution of (3.6). Moreover, for any (quasi-open) solution Ω* of (3.6), uΩ* is locally Lipschitz continuous on D. If D is connected, then all solutions Ω* of (3.6) are open. Remark 3.19. If D is not connected, then it may happen that Ω* is not open: we refer for instance to Example 3.28. However uΩ* is always locally Lipschitz continuous. Let us mention that the existence of an optimal open set for (3.6) had first been proved in [469]. A different penalization was used and it was proved that the corresponding state function converged to a Lipschitz optimal eigenfunction.
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- 56. TENTING TONIGHT ON THE OLD CAMP GROUND. Universal Pictures Co., Inc., c1942. 6 reels, sd. Credits: Associate producer, Oliver Drake; director, Lewis D. Collins; original story, Harry Fraser; screenplay, Elizabeth Beecher; music director, H. J. Salter; photography, William Sickner; film editor, Charles Maynard. © Universal Pictures Co., Inc.; 21Sep42; LP11598. TEORIA DE VOO. Encyclopaedia Britannica Films, Inc., c1946. 1 reel, sd., 16mm. Credits: Collaborators: R. Joseph Stephenson, Walter Brownell. © Encyclopaedia Britannica Films, Inc.; 4Jun46; MP716. THE TERMITE'S LOVE SONG. Soundies Distributing Corp. of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 16Nov42; MP13106. TERMODINÃMICA. Encyclopaedia Britannica Films, Inc., c1946. 1 reel, sd., 16mm. Credits: Collaborator, H. Horton Sheldon. © Encyclopaedia Britannica Films, Inc.; 17Jun46; MP782. TERRA MEXICANA. Encyclopaedia Britannica Films, Inc., c1946. 1 reel, sd., 16mm. Credits: Collaborator, Wallace W. Atwood. © Encyclopaedia Britannica Films, Inc.; 10Jul46; MP892. TERRITORIAL POSSESSIONS OF THE UNITED STATES. International Geographic Pictures, c1939. 2 reels, sd. Credits: Script, Richard Montague; narration, John S. Martin.
- 57. © International Geographic Pictures; 15Jun39; MP9877. TERROR BY NIGHT. Universal Pictures Co., Inc., c1946. 6 reels, sd. Adapted from a story by Sir Arthur Conan Doyle. Credits: Producer and director, Roy William Neill; screenplay, Frank Gruber; music director, Mark Levant; film editor, Saul A. Goodkind. © Universal Pictures Co., Inc.; 8Feb46; LP182. TERROR TRAIL. Columbia Pictures Corp., c1946. 6 reels, sd. Credits: Producer, Colbert Clark; director, Ray Nazarro; original story and screenplay, Ed Earl Repp. © Columbia Pictures Corp.; 21Nov46; LP676. TERRORS ON HORSEBACK. c1946. Presented by P.R.C. Pictures, Inc. 6 reels, sd., 35mm. Credits: Producer, Sigmund Neufeld; director, Sam Newfield; original story and screenplay, George Milton; music director, Lee Zahler; film editor, Holbrook N. Todd. © Pathe Industries, Inc.; 14Aug46; LP491. TERRY AND THE PIRATES. Columbia Pictures Corp., c1940. 2 reels each (no. 1, 3 reels), sd. Based upon the cartoon strip created by Milton Caniff. © Columbia Pictures Corp. 1. Into the Great Unknown. © 3Apr40; LP9525. 2. The Fang Strikes. © 6Apr40; LP9540. 3. The Mountain of Death. © 13Apr40; LP9565. 4. The Dragon Queen Threatens. © 20Apr40; LP9619. 5. At the Mercy of a Mob. © 27Apr40; LP9671. 6. The Scroll of Wealth. © 4May40; LP9672. 7. Angry Waters. © 11May40; LP9681. 8. The Tomb of Peril. © 18May40; LP9682.
- 58. 9. Jungle Hurricane. © 28May40; LP9705. 10. Too Many Enemies. © 4Jun40; LP9695. 11. Walls of Doom. © 12Jun40; LP9696. 12. No Escape. © 18Jun40; LP9713. 13. The Fatal Mistake. © 22Jun40; LP9736. 14. Pyre of Death. © 29Jun40; LP9747. 15. The Secret of the Temple. © 6Jul40; LP9759. A TEST OF SLUDGE SOLVENTS. Brilco Laboratories. 400 ft. Summary: A test of sludge solvents conducted by Foster D. Snell, Inc. Advertises Brilco Sludge Solvent. © Brilco Laboratories; title, descr., 6 prints, 10Mar49; MU3845. TEST TUBE TALE. © Jam Handy Picture Service, Inc.; title, descr., 682 prints; 20Mar41; MU10950. TESTING THE EXPERTS. Paramount Pictures Inc., c1946. 1 reel, sd., 35mm. (Grantland Rice Sportlight) Credits: Narrator, Ted Husing. © Paramount Pictures Inc.; 12Apr46; MP449. TEX BENEKE AND HIS ORCHESTRA. Universal Pictures Co., Inc., c1948. 15 min., sd., bw, 35mm. Summary: A musical short. Credits: Producer and director, Will Cowan; film editor, Ralph Dawson. © Universal Pictures Co., Inc.; 23Mar48; MP2872. TEX BENEKE AND THE GLENN MILLER ORCHESTRA. Universal Pictures Co., Inc., c1946. 2 reels, sd., bw, 35mm.
- 59. Credits: Director, Will Cowan; music director, Milton Rosen. © Universal Pictures Co., Inc.; 18Dec46; LP756. TEX GRANGER. Columbia Pictures Corp., c1948. 2 reels each (no. 1, 3 reels), sd., bw, 35mm. Based on the Tex Granger adventures featured in Calling All Boys and Tex Granger comic magazines. © Columbia Pictures Corp. Credits: Producer, Sam Katzman; director, Derwin Abrahams; story, George H. Plympton; screenplay, Arthur Hoerl, Lewis Clay, Harry Fraser, Royal Cole; film editor, Earl Turner. 1. Tex Finds Trouble. © 1Apr48; LP1620. 2. Rider of Mystery Mesa. © 4Apr48 (in notice: 1947); LP1549. 3. Dead or Alive. © 15Apr48 (in notice: 1947); LP1552. 4. Dangerous Trails. © 22Apr48; LP1562. 5. Renegade Pass. © 29Apr48 (in notice: 1947); LP1573. 6. A Crooked Deal. © 6May48; LP1585. 7. The Rider Unmasked. © 13May48; LP1587. 8. Mystery of the Silver Ghost. © 20May48 (in notice: 1947); LP1603. 9. The Rider Trapped. © 27May48; LP1632. 10. Midnight Ambush. © 3Jun48; LP1639. 11. Renegade Roundup. © 10Jun48; LP1647. 12. Carson's Last Draw. © 17Jun48 (in notice: 1947); LP1658. 13. Blaze Takes Over. © 24Jun48; LP1819. 14. Riding Wild. © 1Jul48 (in notice: 1947); LP1695. 15. The Rider Meets Blaze. © 8Jul48 (in notice: 1947); LP1704. TEX WILLIAMS AND HIS WESTERN CARAVAN. Universal Pictures Co., Inc., c1948. 2 reels, sd., bw, 35mm. Summary: A musical short. Credits: Producer and director, Will Cowan; film editor, Frank Gross.
- 60. © Universal Pictures Co., Inc.; 23Mar48; (in notice: 1947); MP2870. TEXAS. Columbia Pictures Corp., c1941. 10 reels, sd. Credits: Producer, Samuel Bischoff; director, George Marshall; story, Michael Blankfort, Lewis Meltzer; screenplay, Horace McCoy, Lewis Meltzer, Michael Blankfort; music director, M. W. Stoloff; film editor, William Lyon. © Columbia Pictures Corp.; 9Oct41; LP10756. TEXAS. Time, Inc., c1944. 2 reels. © Time, Inc.; 2Oct44; MP15215. TEXAS, BROOKLYN, AND HEAVEN. Golden Productions, Inc. Released through United Artists, Inc., c1948. 89 min., sd., bw, 35mm. Based on the story by Barry Benefield. Summary: A comedy in which a young man and woman from Texas meet on the way to New York. The young woman's involvement with a reformed pickpocket and three spinsters results in absurd business ventures, such as an attempt to put an insolvent riding academy, equipped with mechanical animals, on its feet. Credits: Producer, Robert S. Golden; director, William Castle; screenplay, Lewis Meltzer; music director, Emil Newman; music score, Arthur Lange; film editor, James Newcom. Cast: Guy Madison, Diana Lynn, James Dunn, Lionel Stander, Florence Bates. © Golden Productions, Inc.; 27Aug48; LP1796. TEXAS HOME. Soundies Distributing Corp. of America, Inc., c1945. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 3Sep45; MP16262.
- 61. THE TEXAS KID. Monogram Pictures Corp., c1943. 6 reels, sd. Credits: Producer, Scott R. Dunlap; director, Lambert Hillyer; story, Lynton Wright Brent; screenplay, Jess Bowers; music director, Edward Kay; photography, Harry Neumann; film editor, Carl Pierson. © Monogram Pictures Corp.; 15Oct43; LP12391. TEXAS MANHUNT. P.R.C. Pictures, Inc., c1942. 6 reels, sd. Credits: Producer, Sigmund Neufeld; director, Peter Stewart; original screenplay, William Lively; music, Johnny Lange, Lew Porter; film editor, Holbrook N. Todd. © P.R.C. Pictures, Inc.; 8Dec42; LP13689. THE TEXAS MARSHAL. Producers Releasing Corp., c1941. 6 reels, sd. Credits: Producer, Sigmund Neufeld; director, Peter Stewart; original screenplay, William Lively; film editor, Holbrook N. Todd. © Producers Releasing Corp.; 26May41; LP10486. TEXAS MASQUERADE. Released through United Artists, c1943. Presented by Harry Sherman Productions. 58 min., sd. Based on characters created by Clarence E. Mulford. Credits: Producer, Harry Sherman; director, George Archainbaud; screenplay, Norman Houston, Jack Lait, Jr.; music director, Irvin Talbot; film editor, Walter Hannemann. © United Artists Productions, Inc.; 8Dec43; LP12523. TEXAS PANHANDLE. Columbia Pictures Corp., c1945. 6 reels, sd. Credits: Producer, Colbert Clark; director, Ray Nazarro; original screenplay, Ed. Earl Repp. © Columbia Pictures Corp.; 20Dec45; LP58.
- 62. TEXAS RANGERS RIDE AGAIN. Paramount Pictures Inc., c1940. 7 reels, sd. Credits: Director, James Hogan; original story and screenplay, William R. Lipman, Horace McCoy; photographer, Archie Stout; film editor, Arthur Schmidt. © Paramount Pictures Inc.; 13Dec40; LP10124. TEXAS REDHEADS. RKO Pathe, Inc., c1948. 8 min., sd., bw, 35mm. (Sportscope, no. 1) Summary: As thousands of ducks, including the Texas Redheads, migrate to the lower Rio Grande Valley, Eltinge Warner, publisher of Field and Stream Magazine, and Robert Montgomery, international sportsman, enjoy a successful day hunting. Credits: Producer, Jay Bonafield; director, Joseph Walsh; narrator, Andre Baruch; music, Nathaniel Shilkret; editor, David Cooper. © RKO Pathe, Inc.; 24Sep48; MP3488. TEXAS STAGECOACH. Columbia Pictures Corp., c1940. 6 reels. Credits: Director, Joseph H. Lewis; original screenplay, Fred Myton. © Columbia Pictures Corp.; 13May40; LP9640. THE TEXAS STRIP. Soundies Distributing Corp. of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 16Nov42; MP13107. TEXAS TERRORS. c1940. Presented by Republic Pictures. 6 reels, sd. Credits: Associate producer and director, George Sherman; original screenplay, Doris Schroeder, Anthony Coldewey; music
- 63. director, Cy Feuer; photographer, John MacBurnie; film editor, Tony Martinelli. Appl. author: Republic Productions, Inc. © Republic Pictures Corp.; 22Nov40; LP10094. TEXAS TO BATAAN. Range Busters, Inc., c1942. 7 reels, sd. Credits: Producer, George W. Weeks; director, Robert Tansey; story and screen adaptation, Arthur Hoerl; music direction, Frank Sanucci; photography, Robert Cline; film editor, Roy Claire. © Range Busters, Inc.; 11Sep42; LP11592. TEXAS TROUBLE SHOOTERS. Range Busters, Inc., c1942. 6 reels, sd. (The Range Busters, no. 15) Credits: Producer, George W. Weeks; director, S. Roy Luby; story, Elizabeth Beecher; screen adaptation, Arthur Hoerl; music direction, Frank Sanucci; photography, Robert Cline; film editor, Roy Claire. © Range Busters, Inc.; 12Jun42; LP11417. THANK YOUR LUCKY STARS. Warner Bros. Pictures, Inc., c1943. 127 min., sd. A Warner Bros.-First National picture. From an original story by Everett Freeman and Arthur Schwartz. Credits: Producer, Mark Hellinger; director, David Butler; screenplay, Norman Panama, Melvin Frank, James V. Kern; director, Leo F. Forbstein; orchestral arrangements, Ray Heindorf; film editor, Irene Morra. © Warner Bros. Pictures, Inc.; 9Oct43; LP12303. THANK YOUR LUCKY STARS AND STRIPES. Soundies Distributing Corp. of America, Inc., c1941. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 15Sep41; MP11572.
- 64. THANKFUL DANDELION. C. O. Baptista Films, c1947. 14 min., sd., bw, and color, 16mm. © C. O. Baptista Films, owner of Scriptures Visualized Institute, 15Feb47; MP2247. THANKS FOR THE BOOGIE RIDE. Soundies Distributing Corp of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 9Feb42; MP12180. THAR SHE COMES! Soundies Distributing Corp. of America, Inc., c1944. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 8May44; MP14809. THAT BABIES MAY LIVE. Apex Film Corp., c1949. Presented by the Carnation Co. 47 min., sd., bw, 16mm. Summary: Shows how doctors and scientists have discovered the importance of formulas in infant feeding. Emphasizes that evaporated milk increases a baby's chance of survival through its first year. Credits: Producer, Jack Chertok; director, Sammy Lee; screenplay, David P. Sheppard; narrator, Gerald Mohr; music, David Chudnow; film editor, Frank Capacchione. Cast: Heather Angel, Frederick Worlock, Jimmie Clark, Art Baker, Donald Woods. © Apex Film Corp.; 1Oct49; MP4801. THAT BRENNAN GIRL. Republic Pictures Corp., c1946. 95 min., sd., bw, 35mm. Based on the story by Adela Rogers St. Johns. Credits: Producer and director, Alfred Santell; screenplay, Doris Anderson; music score, George Antheil; music director, Cy Feuer; film director, Arthur Roberts.
- 65. Cast: James Dunn, Mona Freeman, William Marshall, June Duprez. Appl. author: Republic Productions, Inc. © Republic Pictures Corp.; 11Dec46; LP745. THAT DID IT, MARIE. Soundies Distributing Corp. of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 23Feb42; MP12230. THAT FORSYTE WOMAN. Loew's Inc., c1949. 112 min., sd., color, 35mm. An MGM picture. Based on The Forsyte Saga by John Galsworthy. Summary: The way of life of a self-centered family clan is disrupted when an outsider marries into the family. Setting, London in the 1880s. Credits: Producer, Leon Gordon; director, Compton Bennett; screenplay, Jan Lustig, Ivan Tors, James B. Williams; music score, Bronislau Kaper; film editor, Frederick Y. Smith. Cast: Errol Flynn, Greer Garson, Walter Pidgeon, Robert Young, Janet Leigh. © Loew's Inc.; 20Oct49; LP2596. THAT GAL SALOMAY. Soundies Distributing Corp. of America, Inc., c1941. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 4Aug41; MP11382. THAT GANG OF MINE. Monogram Pictures Corp., c1940. 7 reels, sd. Credits: Producer, Sam Katzman; director, Joseph Lewis; original story, Alan Whitman; screenplay, William Lively; photography, Bob Cline; film editor, Carl Pierson.
- 66. © Monogram Pictures Corp.; 24Sep40; LP9950. THAT HAGEN GIRL. Warner Bros. Pictures, Inc., c1947. 83 min., sd., bw, 35mm. From the novel by Edith Kneipple Roberts. Credits: Producer, Alex Gottlieb; director, Peter Godfrey; screenplay, Charles Hoffman; music, Franz Waxman; music director, Leo F. Forbstein; orchestral arrangement, Leonid Raab; film editor, David Weisbart. Cast: Ronald Reagan, Shirley Temple, Rory Calhoun, Lois Maxwell, Dorothy Peterson. © Warner Bros. Pictures, Inc.; 1Nov47; LP1282. THAT HAMILTON WOMAN. Released thru United Artists, c1941. Presented by Alexander Korda; original screenplay, Walter Reisch, R. C. Sherriff; music, Miklos Rozsa; cinematographer, Rudolph Mate; film editor, William Hornbeck. © Alexander Korda Films, Inc.; 27Mar41; LP10361. THAT LADY IN ERMINE. Twentieth Century-Fox Film Corp., c1948. 89 min., sd., color, 35mm. Summary: A light comedy in which the princess of a small mid-European kingdom saves her country by charming the commander of an invading army. Credits: Producer and director, Ernst Lubitsch; screenplay, Samson Raphaelson; music director, Alfred Newman; editor, Dorothy Spencer. Cast: Betty Grable, Douglas Fairbanks, Jr., Cesar Romero, Walter Abel, Reginald Gardiner. © Twentieth Century-Fox Film Corp.; 10Aug48; LP2027. THAT MAN OF MINE. Distributed by Soundies Distributing Corp. of America, Inc., c1946. 1 reel, sd. An Alexander production.
- 67. © Soundies Distributing Corp. of America, Inc.; 23Sep46; MP1244. THAT MIDNIGHT KISS. Loew's Inc., c1949. 99 min., sd., color, 35mm. An MGM picture. Summary: The musical romance of two young opera-singers. Credits: Producer, Joe Pasternak; director, Norman Taurog; screenplay, Bruce Manning, Tamara Hovey; music director, Charles Previn; film editor, Gene Ruggiero. Cast: Kathryn Grayson, Jose Iturbi, Ethel Barrymore, Keenan Wynn, Mario Lanza. © Loew's Inc.; 23Aug49; LP2504. THAT NIGHT IN MANHATTAN. Soundies Distributing Corp. of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 9Nov42; MP13103. THAT NIGHT IN RIO. Twentieth Century-Fox Film Corp., c1941. 8,175 ft., sd. Based on a play by Rudolph Lothar and Hans Adler. Credits: Director, Irving Cummings; screenplay, George Seaton, Bess Meredyth, Hal Long; adaptation, Jessie Ernst; music director, Alfred Newman. © Twentieth Century-Fox Film Corp.; 11Apr41; LP10399. THAT NIGHT WITH YOU. Universal Pictures Co., Inc., c1945. 84 min., sd. Based on a story by Arnold Belgard. Credits: Director, William A. Seiter; screenplay, Michael Fessier, Ernest Pagano; music director, H. J. Salter; music adapted by H. J. Salter, Edward Ward; film editor, Fred R. Feitshans, Jr. © Universal Pictures Co., Inc.; 21Sep45; LP13496.
- 68. THAT OL' GHOST TRAIN. Soundies Distributing Corp. of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 1Jun42; MP12637. THAT OTHER WOMAN. Twentieth Century-Fox Film Corp., c1942. 6,852 ft., sd. Credits: Director, Ray McCarey; screenplay, Jack Jungmeyer, Jr.; music direction, Cyril J. Mockridge. © Twentieth Century-Fox Film Corp.; 13Nov42; LP11771. THAT ROOTIN' TOOTIN', SHOOTIN' MAN FROM TEXAS. Soundies Distributing Corp. of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 13Jul42; MP12778. THAT SPOT. SEE Sign of the Wolf. THAT TEXAS JAMBOREE. Columbia Pictures Corp., c1946. 7 reels, sd. Credits: Producer, Colbert Clark; director, Ray Nazarro; story, Paul Gangelin; screenplay, J. Benton Cheney; music direction, Mischa Bakaleinikoff. © Columbia Pictures Corp.; 16May46; LP374. THAT UNCERTAIN FEELING. Released through United Artists, c1941. Presented by Victorien Sardou and Emile de Najac. Credits: Producer and director, Ernst Lubitsch; screenplay, Donald Ogden Stewart; adaptation, Walter Reisch; music, Werner Heymann; film editor, William Shea. © Ernst Lubitsch Productions, Inc.; 21Feb41; LP10301.
- 69. THAT WAY WITH WOMEN. Warner Bros. Pictures, Inc., c1947. 84 min., sd., bw, 35mm. A first National picture. From a story by Earl Derr Biggers. Credits: Producer, Charles Hoffman; director, Frederick de Cordova; screenplay, Leo Townsend; music, Frederick Hollander; music director, Leo F. Forbstein; orchestral arrangements, Leonid Raab; film editor, Folmer Blangsted. Cast: Dane Clark, Martha Vickers, Sydney Greenstreet, Alan Hale, Craig Stevens. © Warner Bros. Pictures, Inc.; 29Mar47; LP897. THAT WONDERFUL URGE. Twentieth Century-Fox Film Corp., c1948. 82 min., sd., bw, 35mm. A new version of the 1937 motion picture Love Is News. Based on a story by William R. Lipman and Frederick Stephani. Summary: When a tabloid reporter writes lurid stories about the private life of an heiress, she retaliates by announcing her marriage to the reporter. Credits: Producer, Fred Kohlmar; director, Robert Sinclair; screenplay, Jay Dratler; music director, Lionel Newman; editor, Louis Loeffler. Cast: Tyrone Power, Gene Tierney, Reginald Gardiner, Arleen Whelan, Lucile Watson. © Twentieth Century-Fox Film Corp.; 21Dec48; LP2125. THAT WONDERFUL, WORRISOME FEELING. Soundies Distributing Corp. of America, Inc., c1944. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 10Apr44; MP14703. THAT'S A LOTTA SCHICKLGRUBER. Soundies Distributing Corp. of America, Inc., c1943. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 19Jul43; MP13741.
- 70. THAT'S ALL BROTHER, THAT'S ALL. Distributed by Soundies Distributing Corp. of America, Inc., c1946. Presented by R. C. M. Productions, Inc. 1 reel, sd. Credits: Producer, Ben Hersh; director, Dave Gould. © Soundies Distributing Corp. of America, Inc.; 10Nov46; MP1294. THAT'S AN IRISH LULLABY. Soundies Distributing Corp. of America, Inc., c1945. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 19Feb45; MP15631. THAT'S FOR ME. Soundies Distributing Corp. of America, Inc., c1941. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 29Dec41; MP11997. THAT'S HOW I SPELL IRELAND. Soundies Distributing Corp. of America, Inc., c1941. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 28Jul41; MP11424. THAT'S IRELAND. Soundies Distributing Corp. of America, Inc., c1943. © Soundies Distributing Corp. of America, Inc., 31Dec43; MP14447. THAT'S MY BABY. c1944. Presented by Republic Pictures. 7 reels, sd. Credits: Producer, Walter Colmes; director, William Berke; original story, Irving Wallace; screenplay, Nicholas Barrows, William Tunberg; music director, Jay Chernis; photographer, Robert Pittack; film editor, Robert Johns.
- 71. Appl. author: Republic Productions, Inc. © Republic Pictures Corp.; 1Sep44; LP12829. THAT'S MY GAL. Republic Productions, Inc., c1947. 66 min., sd., color, 35mm. Credits: Associate producer, Armand Schaefer; director, George Blair; original story, Frances Hyland, Bernard Feins; screenplay, Joseph Hoffman; music director, Morton Scott; film editor, Arthur Roberts. Cast: Lynne Roberts, Donald Barry, Pinky Lee, Frank Jenks, Jan Savitt. © Republic Pictures Corp.; 7May47; LP1064. THAT'S MY MAN. Republic Productions, Inc., c1947. 104 min., sd., bw, 35mm. Credits: Producer and director, Frank Borzage; written by Steve Fisher, Bradley King; music score, Hans Salter; music director, Cy Feuer; film editor, Richard L. Van Enger. Cast: Don Ameche, Catherine McLeod, Roscoe Karns, John Ridgely, Joe Hernandez. © Republic Pictures Corp.; 7May47; LP1034. THAT'S MY WEAKNESS NOW. Soundies Distributing Corp. of America, Inc., c1941. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 17Nov41; MP11779. THAT'S RIGHT—YOU'RE WRONG. RKO Radio Pictures, Inc., c1939. 95 min., sd. Credits: Producer and director, David Butler; story, David Butler, William Conselman; screenplay, William Conselman, James V. Kern; music arrangements, George Duning; editor, Irene Morra.
- 72. © RKO Radio Pictures, Inc.; 24Dec39; LP9386. THAT'S THE HAWAIIAN IN ME. Soundies Distributing Corp. of America, Inc., c1945. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 15Oct45; MP16395. THAT'S THE MOON. Soundies Distributing Corp. of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 31Dec42; MP13192. THAT'S THE SPIRIT. Universal Pictures Co., Inc., c1945. 10 reels, sd. Credits: Associate producers, Michael Fessier, Ernest Pagano; director, Charles Lamont; original screenplay, Michael Fessier, Ernest Pagano; photography, Charles Van Enger; film editor, Fred Feitshans. © Universal Pictures Co., Inc.; 24Apr45; LP13254. THAT'S WHAT I LIKE ABOUT SWING (CORN). Soundies Distributing Corp. of America, Inc., c1944. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 27Mar44; MP14662. THAT'S WHY I LEFT YOU. Loew's Inc., c1943. Presented by Metro-Goldwyn-Mayer. 862 ft., sd., bw. (John Nesbitt's Passing Parade) Credits: Director, Edward Cahn; original story, Doane Hoag; music score, Max Terr, Nathaniel Shilkret; film editor, Joseph Dietrick. © Loew's Inc.; 8Jun43; LP12134.
- 73. THEIR DIZZY DAY. Warner Bros. Pictures, Inc., c1944. 10 min., sd. (Vitaphone Varieties) Credits: Director and photographer, Mervyn Freeman; narration, Roger Q. Denny; narrator, George O'Hanlon. © Warner Bros. Pictures, Inc.; 9Oct44; MP15290. THEN AND NOW. Warner Bros. Pictures, Inc., c1941. 10 min., sd. (Hollywood Novelty) Credits: Commentator, Knox Manning. © Warner Bros. Pictures, Inc.; 24Dec41; MP12480. THEN IT ISN'T LOVE. Soundies Distributing Corp. of America, Inc., c1946. 1 reel, sd. Credits: Director, William Forest Crouch. © Soundies Distributing Corp. of America, Inc.; 21Oct46; MP1227. THÉORIE MOLÉCULAIRE DE LA MATIÈRE. Encyclopaedia Films, Inc., c1946. 1 reel, sd., 16mm. © Encyclopaedia Films, Inc.; 6Jun46; MP844. THEORY OF FLIGHT. Erpi Classroom Films, Inc., c1941. 1 reel, sd. © Erpi Classroom Films, Inc.; 12Jun41; MP14225. THERE AIN'T A TOWN IN TEXAS. Soundies Distributing Corp. of America, Inc., c1945. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 26Nov45; MP16567. THERE AIN'T NO SUCH ANIMAL. Warner Bros. Pictures, Inc., c1942. 10 min., sd. (Hollywood Novelty)
- 74. Credits: Narration, Joel Maline, Rich Hall; narrator, Knox Manning. © Warner Bros. Pictures, Inc.; 8May42; MP12448. THERE ARE EIGHTY REASONS WHY. Soundies Distributing Corp. of America, Inc., c1945. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 10Sep45; MP16301. THERE GOES KELLY. Monogram Pictures Corp., c1945. 6 reels, sd. Credits: Associate producer, William Strohbach; director, Phil Karlstein; original screenplay, Edmond Kelso; music director, Edward J. Kay; photographer, William Sickner. © Monogram Pictures Corp.; 15Jan45; LP13127. THERE GOES THAT GUITAR. Soundies Distributing Corp. of America, Inc., c1944. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 4Dec44; MP15460. THERE I GO. Soundies Distributing Corp. of America, Inc., c1941. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 16Feb41; MP10853. THERE I GO. Techniprocess, c1941. 1 reel, sd. Credits: Producer, Mario Castegnaro; written and directed by Roy Mack; music director, Lud Gluskin; photograph, Ralph Hammeras. © Techniprocess Special Effects Corp. d.b.a. Techniprocess; 26Oct41; MP11968.
- 75. THERE IS A TAVERN IN THE TOWN. Soundies Distributing Corp. of America, Inc., c1941. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 1Dec41; MP11876. THERE IS NO SUNSHINE. Soundies Distributing Corp. of America, Inc., c1945. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 12Nov45; MP16536. THERE MUST BE A WAY. Soundies Distributing Corp. of America, Inc., c1945. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 29Oct45; MP16452. THERE WAS A LITTLE GIRL. Soundies Distributing Corp. of America, Inc., c1941. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 21Apr41; MP11080. THERE WAS A LITTLE MAN. SEE The Luck of the Irish. THERE WON'T BE A SHORTAGE OF LOVE. Soundies Distributing Corp. of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 6Jul42; MP12733. THERE'LL ALWAYS BE AN IRELAND—AND THE BLARNEY STONE. Soundies Distributing Corp. of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 30Nov42; MP13122.
- 76. THERE'LL BE SOME CHANGES MADE. Soundies Distributing Corp. of America, Inc., c1941. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 21Jul41; MP11346. THERE'S A HOLE IN THE OLD OAKEN BUCKET. Soundies Distributing Corp. of America, Inc., c1941. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 16Jun41; MP11236. THERE'S A PAMPAS MOON ON THE CAMPUS. Soundies Distributing Corp. of America, Inc., c1942. 1 reel, sd. © Soundies Distributing Corp. of America, Inc.; 13Jul42; MP12781. THERE'S GOOD BOOS TONIGHT. Paramount Pictures Inc., c1948. 9 min., sd., color, 35mm. (Noveltoon) Credits: Director, I. Sparber; story, Bill Turner, Larry Riley; narration, Frank Gallop; animation, Myron Waldman, Morey Reden, Nick Tafuri; music, Winston Sharples. © Paramount Pictures Inc.; 23Apr48 (in notice: 1947); LP1574. THERE'S MONEY IN IT. Jam Handy Organization, Inc. Presented by Dearborn Motors Corp. 40 ft., sd., color, 35mm. Summary: Demonstrates that the Ford tractor is adaptable to all sorts of farm work. © Dearborn Motors Corp.; title, descr., 4 prints, 13Apr49; MU4003. THERE'S MUSIC IN YOUR HAIR. Distributed by Columbia Pictures Corp., c1941. 616 ft., sd. (Phantasy, no. 11)
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