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0DWKHPDWLFV6XEMHFWODVVL¿FDWLRQ[[[['[[](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-7-320.jpg)
![Operator Theory:
Advances and Applications, Vol. 168, 1–26
c
2006 Birkhäuser Verlag Basel/Switzerland
Harnack Inequality and Applications to
Solutions of Biharmonic Equations
Gabriella Caristi and Enzo Mitidieri
This paper is dedicated to our friend Philippe Clément for “not killing birds”
Abstract. We prove Harnack type inequalities for linear biharmonic equa-
tions containing a Kato potential. Various applications to local boundedness,
Hölder continuity and universal estimates of solutions for biharmonic equa-
tions are presented.
1. Introduction
During the last decade considerable attention has been paid to solutions of the
time-independent Schrödinger equation
−∆u = V (x)u, x ∈ Ω ⊆ RN
, (1.1)
where Ω is an open subset of RN
and the potential V belongs to the Kato class
KN,1
loc (Ω). See, e.g., Aizenman and Simon [1], Zhao [28], [29], [30], Fabes and Strook
[8], Chiarenza, Fabes and Garofalo [7], Hinz and Kalf [13], Serrin and Zou [22],
Simader [23] and the references therein. Following different approaches these au-
thors have studied the regularity of the solutions and proved a Harnack-type
inequality. Such inequality in turn can be used to prove results such as strong
maximum principles, removable point singularities, existence of solutions for the
Dirichlet problem, Liouville theorems and universal estimates on solutions for non-
linear equations.
In the present paper we shall discuss weak solutions of the following fourth-
order elliptic equation
∆2
u = V u, x ∈ Ω ⊆ RN
, (1.2)
where the potential V is nonnegative in Ω and belongs to KN,2
loc (Ω) the natural
Kato class of potentials associated to the biharmonic operator. See Definition
2.2 in the next section. We point out that Kato classes of potentials associated
to polyharmonic operators and some generalizations have been used in a series](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-13-320.jpg)
![2 G. Caristi and E. Mitidieri
of works by Bachar et al. see [3], [4] and Maagli et al. [14]. In these papers the
authors prove various interesting results including 3G type Theorems and existence
of positive solutions for second order and semilinear polyharmonic equations.
Here we are interested in several kinds of results of qualitative nature and
mainly some of the consequences that can be deduced from Harnack inequality that
we are going to prove during the course. The first one is the regularity problem
which includes the local boundedness and the Hölder continuity of solutions. The
second kind of results are Harnack-type inequalities. The prototype version of these
states: There exist constants C = C(N) and r 0 depending on Ω and norms of
V such that all solutions of (1.2) with u ≥ 0, −∆u ≥ 0 in Ω (i.e., in the sense of
distributions in Ω) satisfy
sup
Br/2
u(x) ≤ C inf
Br/2
u(x), (1.3)
where Br/2 denotes any ball contained in Ω.
Our interest in these results was stimulated by studying certain nonlinear
biharmonic equations and their isolated singularities (see [5], [25]). For several
questions concerning the nature of non-removable singularities and the behavior
of a positive solution in a neighborhood of the isolated singularity it is customary
to assume that V ∈ Lq
loc (Ω) for q N
4 .
On the other hand the technique for proving Lp
-estimates for all p ∞,
relating the sup u to certain integrals involving the solution of (1.2) as employed by
Serrin [20], [21], Stampacchia [26], Trudinger [27] for second order elliptic equations
and by Mandras [15] in the context of weakly coupled linear elliptic systems, seems
not to be easily extendable for this kind of equations.
In order to derive a local a priori majorization for the sup u, we shall follow
two different approaches. The first one was introduced by Simader in [23] and it
is based on representation formulae of solutions and the 3G theorem of Zhao [30],
while the latter have been used by Chiarenza et al [7] and its main ingredients are
Caccioppoli-type inequalities and maximum principle arguments for the operator
∆2
− V .
We remark that the Harnack inequality (1.3) admits a rather simple proof
in the special case V ≡ 0, that is for the biharmonic equation ∆2
u = 0 in Ω. In
fact, due to the special type of mean value formulas for biharmonic functions (see
formula (3.4)0 of Simader [24])
u(x) =
N + 1
2
1
|BR(x)|
BR(x)
u(y)
(N + 2) − (N + 3)
|y − x|
R
dy
it turns out that for x0 ∈ Ω and B2r(x0) ⊂ Ω
sup
Br/2(x0)
|u(x)| ≤
c(N)
|Br(x0)|
Br(x0)
|u(y)| dy,](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-14-320.jpg)
![Harnack Inequality and Biharmonic Equations 3
where c(N) = 2N−1
(N + 1) (2N + 5). On the other hand both conditions u ≥ 0
and −∆u ≥ 0 imply (see [11])
inf
Br/2(x0)
u(x) ≥
(2/3)N
|Br(x0)|
Br(x0)
u(y) dy,
hence inequality (1.3) follows immediately. We observe that in general there is
no hope of obtaining a Harnack inequality for solutions of (1.2) under the only
assumption that they are nonnegative. A simple example in this direction is given
by u(x) = x2
1 in Ω = B2(0). It is interesting to note that the sign conditions
(−∆)m
u ≥ 0 in Ω, for m = 1, . . . , k − 1, can be already found in the classical
book of Nicolescu [18], p. 16, in the context of polyharmonic equations of the type
∆k
u = 0 in Ω, k ∈ N (see also [2]).
This paper is organized as follows. Section 2 contains few preliminary facts,
the proof of the representation formula (see Lemma 2.6) (following Simader [23])
for weak solutions of the problem,
(−∆)m
u = V u, x ∈ Ω ⊆ RN
, (1.4)
where V ∈ KN,m
loc (Ω), N 2m, m ≥ 2, and some remarks on Green functions
for Schrödinger biharmonic operators. In Section 3 we prove the results on local
boundedness and continuity of solutions of (1.2) and a related Harnack inequality
(1.3), (see Theorem 3.6). Namely, for each p ∈ (0, ∞) there exists a constant
C = C(p) and r 0 depending on Ω and some local norms of V such that
sup
Br/2(x0)
|u(x)| ≤ C
1
|Br(x0)|
Br(x0)
|u(y)|p
dy
1/p
. (1.5)
In Section 4 we briefly use the approach by Chiarenza, Fabes and Garofalo
[7], adapted to fourth order problems of the type (1.2). The main outcome of this
technique is an estimate of the modulus of continuity of the solutions is given in
Theorem 4.12 and the Hölder continuity is established in Theorem 4.13 for a class
of potentials V which includes those that belongs to the Lebesgue spaces Lp
loc (Ω)
for p N
4 . Finally Section 5 is devoted to some consequences of Theorem 4.9. The
first application is another proof of Theorem 3.6 while Theorem 5.2 concerns the
behavior at infinity of solutions u ∈ Lp
RN
. Also, in Theorem 5.1 and Theorem
5.3 we prove respectively a Harnack inequality of the type (1.3) and a modified
version of it in the limiting case V = O
1
|x|4 and Ω = BR(0){0}.
We conclude the paper with an application to universal estimates for a semi-
linear biharmonic equation in a general domain (see Theorem 5.5). This result
can be considered a step towards to the general understanding of the existence of
universal estimates for solutions of elliptic systems containing non-linearities with
growth below the so called first critical hyperbola (see [16], [17]).](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-15-320.jpg)
![Harnack Inequality and Biharmonic Equations 5
Example 2.4. Any V belonging to Lα
loc(Ω) with α N/2m. In fact, in this case if
Bt(x) ⊂⊂ Ω we have:
Bt(x)
V (y)
|x − y|N−2m
dy
≤
Bt(x)
|V (y)|α
dy
1
α
Bt(x)
1
|x − y|(N−2m)α dy
1
α
≤ σ
1
α
N V Lα(Bt(x))t2m− N
α . (2.1)
Therefore, Definition 2.2 is satisfied with φV (t) = C(V ) t2m− N
α for t 0.
Definition 2.5. We say that u ∈ L1
loc(Ω) is a distributional solution of (1.4),
or that u is a solution in the sense of distributions, if u ∈ L1
loc(Ω) and for any
ψ ∈ C∞
0 (Ω) we have
Ω
u(y)∆m
ψ(y)dy =
Ω
V (y)u(y)ψ(y)dy (2.2)
For any m ≥ 1, N 2m and r 0 we set
Φm,N (r) = Cm,N r2m−N
, (2.3)
where
Cm,N =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Γ(2 − N/2)
22m−2(m − 1)!Γ(m + 1 − N/2)
, if N is odd,
(−1)m−1 (N/2 − m − 1)!
22m−2(m − 1)!(N/2 − 2)!
, if N is even.
The function Φm,N as function of r = |x|, for x ∈ RN
{0}, is polyharmonic of
degree m and is called the fundamental solution of the equation ∆m
u = 0. Here,
the constants Cm,N are chosen so that ∆p
Φm,N = Φm−p,N for p = 1, . . . , m − 1,
(see [2]). In the sequel, sometimes we will write Φm(x, y) instead of Φm,N (|y − x|).
The following representation formula holds:
Lemma 2.6. Let N 2m. If u ∈ C∞
0 (RN
), then for any x ∈ supp(u)
u(x) = −ΩN
RN
Φm,N (|y − x|)∆m
u(y)dy, (2.4)
where ΩN = ((N − 2)ωN )−1
and ωN = 2πN/2
/Γ(N/2) is the area of the unit
sphere.](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-17-320.jpg)
![Harnack Inequality and Biharmonic Equations 7
Lemma 2.8. Let Ω ⊆ RN
be bounded, ρ ∈ C0
0 (Ω), g ∈ L1
loc(Ω) and 0 α N.
Then, if for any n ∈ N, n 0 and n → 0, as n → ∞, there exists a subsequence,
which we still denote by n, such that
Ω
gn (y)ρ(y)|y − x|−α
dy →
Ω
g(y)ρ(y)|y − x|−α
dy, a.e. in RN
.
We refer to [23] for its proof.
2.1. Green functions of Schrödinger biharmonic operators
We recall some properties of the Green functions associated to the following bound-
ary value problems for the biharmonic equation on the ball Br(0) ⊂ RN
:
∆2
u(x) = f(x), x ∈ Br(0) : (2.10)
that is, the Navier boundary value problem (N):
u = ∆u = 0 on ∂Br(0), (2.11)
and the Dirichlet boundary value problem (D):
u =
∂u
∂ν
= 0 on ∂Br(0). (2.12)
It is well known that both problems admit a nonnegative Green function on Br(0),
which we denote respectively by GN
r (x, y) and GD
r (x, y). Moreover, the following
estimates hold:
Lemma 2.9 (3G-Lemma). Let G = GN
r or GD
r . Then, there exists a constant C1 0
depending only on the dimension N such that for any ball B of RN
we have
G(x, z)G(z, y)
G(x, y)
≤ C1
|x − z|4−N
+ |z − y|4−N
,
for all x, y, z ∈ B.
The proof of this lemma for the Dirichlet problem is contained in [12], while
for the Navier problem it can be straightforwardly obtained by iteration of the
3G-Lemma for the Laplace operator, see [30].
Now, consider instead of (2.10) the Schrödinger biharmonic equation:
∆2
u(x) = V (x)u(x) + f(x), for x ∈ Br(0), (2.13)
where V belongs to the natural Kato class KN,2
loc (Br(0)) associated to ∆2
. The
following result extends Lemma 2.3 of [6] to the biharmonic case.
Proposition 2.10. Assume that V ∈ KN,2
loc (Br(0)). Then, there exists r1 0 such
that if 0 r r1, the problems (2.13)–(2.11) and (2.13)–(2.12) admit a nonneg-
ative Green function on Br(0).](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-19-320.jpg)
![Harnack Inequality and Biharmonic Equations 9
By a similar argument, it can also be proved that if we fix the radius of the ball, for
instance, equal to 1, then, there exists a nonnegative Green function for problems
(2.10)–(2.11) and (2.10)–(2.12) if φV (1, B1) is sufficiently small.
3. Local boundedness and continuity of solutions
In this section we will prove the local boundedness and the continuity of distribu-
tional solutions of problem (2.6), extending in this way Theorem 2.4 of [23]. The
method of proofs is essentially the same.
First of all, we choose η ∈ C∞
0 (R) such that 0 ≤ η(t) ≤ 1, η(t) = 1 if
t ∈ [−1/2, 1/2] and η(t) = 0 if |t| 1. Given δ 0, we set ηδ(z) = η(δ−1
|z|), for
z ∈ RN
.
Lemma 3.1. Assume that V ∈ KN,2
loc (Ω). Let x ∈ Ω and 0 δ 1
4 dist(x, ∂Ω).
Then, for any x = z
Ω
|V (y)|ηδ(x − y)ηδ(z − y)
|x − y|N−4|y − z|N−4
dy ≤ 2N−3
φV (δ, B3δ(x))
η4δ(x − z)
|x − z|N−4
. (3.1)
Proof. First of all, we observe that if σ = |x − z| 2δ, then it follows that
ηδ(x − y)ηδ(z − y) = 0. Hence, we can assume that |x − z| 2δ. Consequently,
σ/2 ≤ δ.
Define Ω1 = {y ∈ Ω : |y − x| ≤ σ/2} and Ω2 = {y ∈ Ω : |y − x| ≥ σ/2},
and denote the corresponding integrals by I1 and I2. For y ∈ Ω1 we have that
|z − y| ≥ |z − x| − |x − y| ≥ σ/2 and therefore
I1 ≤
Ω1
2
σ
N−4
|V (y)|ηδ(x − y)
|x − y|N−4
dy ≤
2
σ
N−4
φV (δ, B3δ(x)).
Similarly,
I2 ≤
Ω2
2
σ
N−4
|V (y)|ηδ(y − z)
|y − z|N−4
dy ≤
2
σ
N−4
φV (δ, B3δ(x)).
Since η4δ(|x − z|) = 1 for σ ≤ 2δ, these inequalities imply (3.1).
Lemma 3.2. Assume that V ∈ KN,2
loc (Ω) and that u is a solution of (2.6) in the
sense of distributions. Let x0 ∈ Ω, R0 0 be such that B2R0 (x0) ⊂⊂ Ω. Choose
0 δ0 R0/4 such that
φV (δ) ≡ φV (δ, BR0 (x0)) 22−N
cN (3.2)
for 0 δ ≤ δ0. Then, for 0 R ≤ R0/2, 0 δ ≤ δ0 and for a.e. z ∈ BR(x0) we
have:
Ω
|V (x)||u(x)|ηδ (x − z)
|x − z|N−4
dx
≤ 2N−4
δ4−N
V u L1(BR+4δ(x0)) + 2Cδ−N
u L1(BR+2δ(x0)), (3.3)
where C depends on N and the choice of η.](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-21-320.jpg)
![10 G. Caristi and E. Mitidieri
Proof. Let x ∈ BR0 (x0) and 0 δ0 R0/4. Since ρ(·) = ηδ(| · −x|) ∈ C∞
0 (Ω), we
can apply the representation formula of Lemma 2.7 and obtain that:
cN u(x) = −
Φ2(x, y)V (y)u(y)ηδ(y − x)dy
+ 2
u(y)(∇∆yηδ(y − x), ∇Φ2(x, y))dy
+ 2
∆2
yηδ(y − x) u(y)Φ2(x, y)dy
+ 2
∆(∇Φ2(x, y), ∇yηδ(y − x))u(y)dy
+
(∇yηδ(y − x), ∇∆Φ2(x, y))u(y)dy
+ 2
u(y)∆Φ2(x, y)∆yηδ(y − x)dy
−
u(y)∆2
ηδ(y − x)Φ2(x, y)dy. (3.4)
Taking into account of the definition of ηδ and of Φ2 and denoting by χ the
characteristic function of the interval [1/2, 1], we get that for any x ∈ BR(x0)
|u(x)| ≤ c−1
N
Φ2(x, y)|V (y)||u(y)|ηδ(y − x)dy
+ Cδ−N
|u(y)|χ(δ−1
|y − x|)dy, (3.5)
where C = C(N, η) 0. Using (3.5) in the left-hand side of (3.3), we obtain
|V (x)||u(x)|ηδ(x − z)
|x − z|N−4
dx
≤ c−1
N
|V (x)||V (y)||u(y)|ηδ(y − x)ηδ(x − z)
|x − z|N−4|x − y|N−4
dxdy
+ Cδ−N
|V (x)|ηδ(x − z)
|x − z|N−4
|u(y)|χ(δ−1
|y − x|)dxdy. (3.6)
Now, we proceed as in the proof of Lemma 2.3 of [23]. The last integral in (3.6)
can be estimated by φV (δ)|u|L1(BR+2δ(x0)). Then, we apply Lemma 2.6 to the first
integral and get:
c−1
N
|V (y)||u(y)|
|V (x)|ηδ(y − x)ηδ(x − z)
|x − z|N−4|x − y|N−4
dxdy
≤ c−1
N 2N−3
φV (δ, B3δ(x))
|V (y)||u(y)|η4δ(y − z)
|y − z|N−4
dydx. (3.7)
We remark that φV (δ, B3δ(x)) ≤ φV (δ) and that
η4δ(y − z) = ηδ(y − z) + (η4δ(y − z) − ηδ(y − z)),](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-22-320.jpg)
![12 G. Caristi and E. Mitidieri
Proposition 3.5. Assume that V ∈ KN,2
loc (Ω) and that u is a solution of (2.6) in
the sense of distributions. Let x0 ∈ Ω, R0 0 be such that B2R0 (x0) ⊂⊂ Ω. Then,
there exists 0 R1 ≤ R0/2 such that for any 0 R R1 the following estimate
holds
u L∞(BR(x0)) ≤ CR−N
u L1(B2R(x0)), (3.11)
where C depends only on N.
Proof. Let R1 ≤ R0/2 be as in Theorem 2.1. Then, for any 0 R R1:
φV (B2R(x0)) ≤ 2−1
cN . (3.12)
We shall estimate V u L1(B3R/2(x0)) with the u L1(B2R(x0)). To this aim, we take
ρ ∈ C∞
0 (R) such that 0 ≤ ρ ≤ 1, ρ(·) ≡ 1 in [0, 3R/2] and ρ(t) = 0 if t ≥ 2R.
Further, we assume that |ρ(i)
(t)| ≤ t−i
for i = 1, 4 and any t. From Lemma 2.7 we
get that for any x ∈ B2R(x0)
cN |u(x)|ρ(x) =
|V (y)||u(y)|Φ2(x, y)ρ(|x − y|)dy
+
|u(y)||R(x, y)|dy, (3.13)
where R is defined in terms of the derivatives of ρ and of Φ2. Multiplying (3.13)
by |V (x)| and integrating over B ≡ B2R(x0) we get:
cN
B
|V (x)||u(x)|ρ(x)dx
≤
|V (x)|
|x − y|N−4
dx
|V (y)||u(y)|ρ(y)dy
+
B
3R
2 ≤|x−y|≤2R
|u(y)|
|V (x)|
|x − y|N−4
dxdy
≤ φV (B2R(x0))
B
|V (x)||u(x)|ρ(x)dx + φV (B2R(x0))
B
|u(x)|dx. (3.14)
From (3.12) and (3.14) we deduce that:
B
|V (x)||u(x)|ρ(x)dx ≤ C
B
|u(x)|dx, (3.15)
which implies that
|V (u)|L1(B 3R
2
(x0)) ≤ c−1
N |u|L1(B2R(x0)). (3.16)
Now, we apply Theorem 2.1 with δ = R/8. From (3.10) and (3.16), it follows that
for any x ∈ BR(x0):
|u(x)| ≤ 2N−4
δ4−N
|V (u)|L1(B 3R
2
(x0)) + Cδ−N
|u|L1(B2R(x0))
≤ CRN
|u|L1(B2R(x0)), (3.17)
where C = C(N).](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-24-320.jpg)
![Harnack Inequality and Biharmonic Equations 13
3.1. The Harnack inequality
Theorem 3.6. Assume that V ∈ KN,2
loc (Ω). Let u be a nonnegative weak solution of
(2.6) such that −∆u ≥ 0 in Ω. Then there exist C = C(N) and R0 = R0(V ) such
that for each 0 R ≤ R0 and B2R(x0) ⊂⊂ Ω we have
sup
BR/2(x0)
u ≤ C inf
BR/2(x0)
u, (3.18)
where C independent of V and u.
Proof. Since u is nonnegative and satisfies −∆u ≥ 0 in Ω, then we have (see,
e.g., [11])
inf
BR/2(x0)
u(x) ≥ C(N)
1
|BR|
BR(x0)
u(y) dy,
for all R 0 such that B2R ⊂ Ω . Then, the conclusion follows from this inequality
and (3.11).
Remark 3.7. The condition −∆u ≥ 0 in Ω is necessary for the validity of the above
Theorem as the following example shows: consider the function u(x) = x2
1. It is
nonnegative, satisfies ∆2
u = 0 and ∆u = 2, but (3.18) does not hold for x0 and
any R.
4. Local boundedness and continuity of solutions:
an alternative approach
In this section we shall prove The Harnack inequality following the approach of
[7], based on Lp
estimates. Assume that m = 2, N 4 and that V ∈ KN,2
loc (Ω).
First of all, we recall the following result from [19]:
Lemma 4.1. Assume that V ∈ KN,2
loc (Ω). Then, for every 0 there exists a
constant C = C(, V ) such that
Ω
|V | ψ2
≤
Ω
|∆ψ|2
+ C
Ω
|ψ|2
, for all ψ ∈ H2,2
0 (Ω) . (4.1)
For later use we need also the following slightly different inequality. If t 0
s ∈ (0, t), then:
Bt
|V | ψ2
≤
Bt
|∆ψ|2
+
C(, V )
(t − s)4
Bt
|ψ|2
, for all ψ ∈ H2,2
0 (Bt) . (4.2)
When V ∈ L
N
4−δ (Ω), δ ∈ (0, 4), a simple proof of the above embedding property
(4.2) can be obtained as follows. By the embedding
L
2N
N−4 (Ω) ∩ L2
(Ω) → L
2N
N−4+δ (Ω)
and by Sobolev’s inequality
ψ
L
2N
N−4 (Ω)
≤ c(N) ∆ψ L2(Ω) , for all ψ ∈ H2,2
0 (Ω) ,](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-25-320.jpg)
![14 G. Caristi and E. Mitidieri
it follows that for every 0
Ω
|V | ψ2
≤ (t − s)
δ
V
L
N
4−δ (Ω)
(t − s)
−δ
ψ
2
L
2N
N−4+δ (Ω)
≤ (t − s)
δ
V N
4−δ
ψ
2
L
2N
N−4 (Ω)
+ C(δ) (t − s)
−4
− 4−δ
δ ψ
2
L2(Ω) .
Hence, by taking 1 = (t − s)
δ
V
L
N
4−δ (Ω)
we obtain
Ω
|V | ψ2
≤ 1
Ω
|∆ψ|
2
+
C
(t − s)
4
Ω
|ψ|
2
, (4.3)
where C depends on 1, δ and (t − s)
δ
V N/4−δ.
Remark 4.2. As a consequence of (4.1) we deduce that, if u ∈ H2,2
loc (Ω), then,
V u ∈ L1
loc (Ω). In fact, let B2r ⊂ Ω and let ϕ ∈ C∞
0 (B2r) be such that 0 ≤ ϕ ≤ 1,
ϕ ≡ 1 on Br. Since uϕ ∈ H2,2
0 (B2r) and
B2r
|V | |u| ϕ2
≤
B2r∩{|u|≤1}
|V | |u| +
B2r∩{|u|1}
|V | |u| ϕ2
≤
B2r
|V | +
B2r
|V | |uϕ|
2
,
it follows that V u ∈ L1
(Br).
Definition 4.3. We say that u ∈ H2,2
loc (Ω) is a weak solution of (2.6), if for any
ψ ∈ C∞
0 (Ω) we have
Ω
∆u(y)∆ψ(y)dy =
Ω
V (y)u(y)ψ(y)dy. (4.4)
The proof of the local boundedness of any weak solution of (2.6) will be proved
through a sequence of lemmas. The first one states a Caccioppoli-type estimate([9]).
We shall omit its proof
Lemma 4.4. Let u be a weak solution of (2.6). If 0 s t and Bt ⊂ Ω, then there
exists a constant C = C(V ) independent of u such that
Bs
|∆u|2
≤
C
(t − s)4
Bt
|u|2
. (4.5)
Remark 4.5. We note that also the following estimate holds:
Bs
|∇u|2
≤
C
(t − s)2
Bt
u2
. (4.6)
where C = C(φV ).](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-26-320.jpg)
![Harnack Inequality and Biharmonic Equations 15
The following lemma is well known see for instance [7].
Lemma 4.6. Assume that there exist θ ∈ (0, 1), α ≥ 0, a 0 and a real continuous,
non-decreasing and positive function I(·) defined on (0, 1] such that
I(s) ≤ a
1
(t − s)
n I(t)
θ
, (4.7)
holds for all 0 s t ≤ 1. Then, for every s1 0 there exists a constant
C = C(θ, n, s1, a) independent of I such that
I (s1) ≤ C. (4.8)
The next results provide a kind of reversed Hölder inequality.
Lemma 4.7. Let B2r be a ball contained in Ω and let u be a weak solution of (2.6).
Then, there exists a constant C(φV ) such that
1
|Br/2|
Br/2
u2
1/2
≤ C
1
|Br|
Br
|u| . (4.9)
Proof. Without restriction we may assume that the center of Br is the origin. Let
ur(x) = u(rx). Then ur is a solution of ∆2
ur = Vrur in B2(0), where Vr(x) =
r4
V (rx) satisfies
sup
x∈B2
B2
|Vr(y)|
|x − y|N−4
dy ≤ sup
0α4r
sup
w∈Ω
Bα(w)
|V (z)| χΩ(z)
|w − z|N−4
dz. (4.10)
Since V ∈ KN,2
loc (Ω), then the right-hand side of (4.10) is bounded and tends to
zero as r → 0. We may establish Lemma 4.7 by taking r = 1, Ω = B2, where B2
is centered at the origin. We may also assume that
B1
|u| = 1.
Define for 0 ≤ s ≤ 1:
I(s) =
Bs
u2
1/2
.
Since L1
(Ω) ∩ L
2N
N−4 (Ω) → L2
(Ω), we get
I(s) ≤
Bs
|u|
2N
N−4
N−4
2(N+4)
.
By the Sobolev inequality we obtain for 0 ≤ s t ≤ 1
I(s) ≤ a(φV )
1
(t − s)2
I(t)
N
N+4
. (4.11)
The conclusion follows from Lemma 4.6.](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-27-320.jpg)
![16 G. Caristi and E. Mitidieri
Lemma 4.8. Let p ∈ (0, 2), n ≥ 0 and a 0. Let v ∈ L∞
(B1) be nonnegative and
sup
Bs
v2
≤
a
(t − s)n
Bt
v2
(x)dx,
for every 0 s t ≤ 1. Then, there exists a constant C(n, p, a) 0 such that
sup
B1/2
vp
≤ C
B1
vp
(x)dx.
Proof. We may suppose that v ≡ 0. Putting
γ =
B1
vp
(x)dx
−2/p
and I(s) = γ
Bs
v2
, for 0 s 1,
we note that
sup
B1/2
vp
≤ [6n
aI (2/3)]
B1
vp
(x)dx.
In order to complete the proof it is sufficient to bound I (2/3) by a constant
depending only on n, p and a. To this end let α = 2
2−p and 0 s t ≤ 1. We
obtain
I(s)α
= γα
Bs
v2−p
vp
α
≤ γα
sup
Bs
v2
Bs
vp
α
≤
aγα
(t − s)
n
Bs
vp
α
Bt
v2
≤
a
(t − s)
n I(t).
Finally, putting θ = 1/α and s1 = 2/3, the conclusion follows from (4.7) and (4.8).
The following theorem states the local boundedness of weak solutions of (2.6).
Theorem 4.9. Let V ∈ KN,2
loc (Ω). Then, for each p ∈ (0, ∞) there exist C = C(p)
0 and r1 = r1(φV ) 0 such that if u is a weak solution of (2.6), then, for each
0 R ≤ R1 and B2R(x0) ⊂ Ω, we have
sup
BR/2(x0)
|u(x)| ≤ C
1
|Br|
BR(x0)
|u(y)|
p
dy
1/p
. (4.12)
Proof. Without restriction we can assume that x0 = 0. Let G(x, y) be the Green
function of the Dirichlet problem for equation (2.13) on B2R, see Proposition 2.10.
Choose R/2 ≤ s t ≤ R and a function ϕ ∈ C∞
0 (Bt−(t−s)/4) such that 0 ≤ ϕ ≤ 1,
ϕ ≡ 1 on B(t+s)/2, and
Dk
ϕ
≤ c/(t − s)k
, for k = 1, 2. We claim that for almost
all x ∈ B2R the following identity holds:
u(x)ϕ(x) =
B2R
∆yG(x, y) u(y) ∆ϕ(y) dy −
B2R
G(x, y) ∆u(y) ∆ϕ(y) dy
+ 2
B2R
∆yG(x, y) (∇u(y), ∇ϕ(y)) dy − 2
B2R
(∇yG(x, y), ∇ϕ(y))∆u(y) dy.](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-28-320.jpg)
![18 G. Caristi and E. Mitidieri
Since
Bt
G(x, y) dy is bounded, from the above estimates we obtain
sup
Bs
|u(x)| ≤
c
(t − s)5
Bt
u(y)2
dy
1/2
. (4.13)
Finally in virtue of Lemma 4.8 and estimate (4.13) we get the conclusion, for
p ∈ (0, 2):
sup
Br/2
|u(x)| ≤ C(p)
Br
|u(y)|
p
dy
1/p
.
The case p ∈ (2, ∞) follows applying Hölder inequality.
The continuity of weak solutions can be proved as in the previous section.
4.1. The Hölder continuity
The proof of Theorem 4.12 below will be based on two lemmas, which are adap-
tations of the Lemmas 3.1 and 3.2 of [23]. The first one is the following.
Lemma 4.10. Let 0 β 1 and x0, x ∈ RN
, r 0 be such that
λ :=
|x − x0|
r
1−β
≤ 1. (4.14)
Then, there exists a constant c 0 such that for y with |y − x0| ≥ 2 r1−β
|x − x0|
β
we have
1
|y − x|
N−4
−
1
|y − x0|
N−4
≤ c
|x − x0|
r
1−β
1
|y − x0|
N−4
. (4.15)
Proof. Let y ∈ RN
be such that
|y − x0| ≥ 2 r1−β
|x − x0|β
. (4.16)
Since |x − x0| = |x − x0|
β
|x − x0|
1−β
≤ λ/2 |y − x0|, it follows that
|y − x| ≥ |y − x0| − |x − x0| ≥ 1 −
λ
2
|y − x0| .
From this inequality we get
1
|y − x|
N−4
−
1
|y − x0|
N−4
= ||y − x0| − |y − x||
N−5
k=0
|y − x0|
k−N+4
|y − x|
−k−1
≤
|x − x0|
|y − x0|
N−3
N−5
k=0
1
1 − λ/2
k+1
.
Hence, the conclusion follows with c =
N−5
k=0 2k
.](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-30-320.jpg)
![20 G. Caristi and E. Mitidieri
Proof. Without loss of generality we may assume that x0 = 0 and set B4r =
B4r(0). Let ϕ ∈ C∞
0 (B2r), 0 ≤ ϕ ≤ 1 with ϕ ≡ 1 on B3r/2 and
Dk
ϕ
≤ c/r|k|
,
for k = 1, 2. We have
u(x)ϕ(x) =
Φ2(x, y) V (y) u(y) ϕ(y) dy −
Φ2(x, y)∆u(y)∆ϕ(y)dy
+
∆yΦ2(x, y) u(y) ∆ϕ(y) dy − 2
∇yΦ2(x, y) · ∇ϕ(y)∆u(y) dy
+ 2
∆yΦ2(x, y)∇u(y) · ∇ϕ(y) dy =
5
i=1
Ii(x).
Fix β ∈ (0, 1) and let |x| r. From this identity we obtain
u(x) − u(0) =
(Φ2(x, y) − Φ2(0, y) ) V (y) u(y) ϕ(y)
−
(Φ2(x, y) − Φ2(0, y)) ∆u(y)∆ϕ(y)dy
+
(∆yΦ2(x, y) − ∆yΦ2(0, y)) u(y)∆ϕ(y)
− 2
(∇yΦ2(x, y) − ∇yΦ2(0, y)) ∇ϕ(y) ∆u(y)
+ 2
(∆yΦ2(x, y) − ∆yΦ2(0, y)) ∇u(y)∇ϕ(y). (4.19)
Since V u ∈ KN,2
loc (B2r) and φV u(t) ≤ supB2r
|u| · φV (t), we can apply Lemma 4.11
to the first term on the right-hand side of (4.19) and obtain
|I1(x) − I1(0)| ≤
c
|x|
r
1−β
ηV (2r) + 2ηV
3r1−β
|x|
β
sup
B2r
|u| . (4.20)
To handle the other terms of (4.19), let us observe that all the derivatives of ϕ
vanish for |y| ≤ r. Define for t ∈ [0, 1] φ(t) := |y − tx|4−N
, where y ∈ B2r B3r/2
and |x| r. From the mean value theorem and |y − t0x| ≥ |y|−|x| ≥ r/2 it follows
that
φ(1) − φ(0) =
|y − x|
4−N
− |y|
4−N
= φ
(t0)
≤ (N − 4)
|x|
|y − t0x|
N−3
≤ (N − 4) 2N−3 |x|
rN−3
,
where t0 ∈ (0, 1). Therefore,
|Φ2(x, y) − Φ2(0, y)| ≤ c
|x|
rN−3
≤ c
|x|
r
1−β
1
rN−4
. (4.21)](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-32-320.jpg)
![Harnack Inequality and Biharmonic Equations 21
The same argument shows also that
|∇yΦ2(x, y) − ∇yΦ2(0, y)| ≤ c
|x|
r
1−β
1
rN−3
, (4.22)
|∆yΦ2(x, y) − ∆yΦ2(0, y)| ≤ c
|x|
r
1−β
1
rN−2
. (4.23)
Now we can estimate the remaining terms of (4.19), that is |Ik(x) − Ik(0)| for
k = 2, . . . , 5, by using the bounds (4.21), (4.22) and (4.23). We find that
|I2(x) − I2(0)| ≤
c
r2
|Φ2(x, y) − Φ2(0, y)| |∆u(y)| dy
≤ c
|x|
r
1−β
1
rN−2
B2r
|∆u| ≤ c
|x|
r
1−β
r2 1
|B2r|
B2r
|∆u|
2
1/2
,
and hence, by Lemma 4.4
|I2(x) − I2(0)| ≤ c
|x|
r
1−β
1
|B3r|
B3r
|u|
2
1/2
. (4.24)
It is also easy to check that
|I3(x) − I3(0)| ≤ c
|x|
r
1−β
1
|B3r|
B3r
|u|
2
1/2
, (4.25)
and
|I4(x) − I4(0)| + |I5(x) − I5(0)| ≤ c
|x|
r
1−β
1
|B3r|
B3r
|u|2
1/2
.
Finally, this inequality together with the estimates (4.20), (4.24) and (4.25) com-
plete the proof.
In order to conclude that the weak solutions of (2.6) are Hölder continuous
we restrict ourselves to a class of potentials V such that
φV (t) ≤ Mtα
(4.26)
for some α 0. Clearly, this class is not empty since, according to Example 2.3,
we know that if V satisfies a Stummel condition, then (4.26) holds with α = γ/2,
or if V belong to Lp
loc with p N
4 , then (4.26) holds with α = 4 − N
p .
We point out that if N ≤ 3 all weak solutions u ∈ H2,2
loc (Ω) of (2.6) are Hölder
continuous by Sobolev’s embedding.
Theorem 4.13. Let x0 ∈ Ω and B4R(x0) ⊂ Ω. Let u be a weak solution of (1.1).
Suppose that there exist α 0 and a constant MV = M(V, B4R) such that (4.26)
holds. Then, if |x − x0| ≤ R, we have
|u(x) − u(x0)| ≤ c
[MV (3R)
α
+ 1] R− α
1+α sup
B3R(x0)
|u|
|x − x0|
α
1+α
. (4.27)](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-33-320.jpg)
![22 G. Caristi and E. Mitidieri
Proof. Since φV (2r) ≤ MV (2R)
α
and
φV
3r
|x − x0|
R
β
≤ MV (3r)
α |x − x0|
R
αβ
,
the conclusion follows directly from Theorem 4.12 by taking β = 1
1+α .
5. Applications
In this section we shall present some consequences of Theorem 4.9. The first is
another proof of Harnack inequality:
Theorem 5.1. Assume that V ∈ KN,2
(Ω). Let u be a nonnegative weak solution of
(1.1) such that −∆u ≥ 0 in Ω. Then there exist C = C(N) and r0 = r0(ηV ) such
that for each 0 R ≤ R0 and B2R ⊂ Ω we have
sup
BR/2
u ≤ C inf
BR/2
u.
Proof. Since u is nonnegative and satisfies −∆u ≥ 0 in Ω, then we have (see,
e.g., [11])
inf
Br/2
u(x) ≥ C(N)
1
|Br|
Br
u(y) dy,
for all r 0 such that B2r ⊂ Ω. The conclusion follows from this inequality and
Theorem 4.9 with p = 1.
Theorem 5.2. Let p : 1 ≤ p ∞ and Ω = RN
BR(0). If u ∈ Lp
(Ω) ∩ H2,2
loc (Ω) is
a solution of (1.1) with V ∈ KN,2
(Ω) nonnegative then:
lim
|x|→∞
|u (x)| = 0
Proof. Without loss of generality we may assume that u is continuous in Ω. By
Theorem 4.9, it follows that there exists r0 = r0(ηV ) such that for all x : |x| ≥ R+2,
for each r ∈ (0, r0) and 0 s t 1 we have
sup
Bsr(x)
|u(y)|p
≤
C
rN (t − s)
N
Btr(x)
|u(z)|p
dz.
Since u ∈ Lp
(Ω) the result follows.
The next result concerns the limiting case for a Harnack’s theorem near the
possible isolated singularity x = 0 (compare with Theorem 3.1 of [10]).
Theorem 5.3. Let Ω = BR(0) {0} and let V ∈ Cloc (Ω) be nonnegative and such
that for all x ∈ Ω
0 ≤ V (x) ≤
c1
|x|
4 , c1 0.
Let u ∈ H2,2
loc (Ω) be a solution of
∆2
u = V u on Ω,
u ≥ 0, −∆u ≥ 0 in Ω.
(5.1)](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-34-320.jpg)
![Harnack Inequality and Biharmonic Equations 23
Then there exist two constants C and ϑ1 depending on c1 such that for each 0
ϑ ≤ ϑ1 and 0 1
2R we have
sup
≤|x|≤(1+ϑ)
u(x) ≤ C inf
≤|x|≤(1+ϑ)
u(x). (5.2)
Proof. The proof is a slight modification of the proof of Theorem 4.9. Here, we shall
emphasize the dependence of the constants on c1. If x0 ∈ Ω let r0 = 1
4 |x0| so that
B2r0 (x0) ⊂ Ω. We claim that for every p ∈ (0, ∞) there exists positive constants
C = C(p) and ϑ0 depending on c1 such that for all x0 ∈ Ω and 0 r ≤ ϑ0r0
sup
Br/2(x0)
|u(x)| ≤ C
1
|Br(x0)|
Br(x0)
|u(y)|
p
dy
1/p
. (5.3)
To prove (5.3) we fix σ ∈ (0, 4) and note that
sup
x∈Ω
rσ
V
L
N
4−σ (Br(x))
= C1 ∞
where C1 depends on c1. This fact together with the embedding property ( 4.2)
yields the Lemma 3.2 for each Bsr0 (x0) ⊂ Btr0 (x0) ⊂ Ω. Clearly, Lemma 4.7 and
4.8 hold true. We have to bound
sup
x∈B2r(x0)
B2r(x0)
V (y)
|x − y|
N−4
dy,
uniformly with respect to x0 ∈ Ω. Indeed, since
sup
x∈B2r(x0)
B2r(x0)
V (y)
|x − y|
N−4
dy ≤ c rσ
0 V
L
N
4−σ (B2r0 (x0))
rσ
rσ
0
,
then
sup
x∈B2r(x0)
B2r(x0)
V (y)
|x − y|
N−4
dy ≤ δ,
holds, provided that 0 r ≤ ϑ0r0, where ϑ0 = δ/(2cC1). Hence, it suffices to
apply those lemmas in order to obtain (5.3). Combining (5.3) with
inf
Br/2(x0)
u(x) ≥
c
|Br(x0)|
Br(x0)
u(y) dy,
corresponding to u ≥ 0 and −∆u ≥ 0, we obtain for 0 r ≤ ϑ0r0
sup
Br/2(x0)
u(x) ≤ C inf
Br/2(x0)
u(x). (5.4)
Finally, the inequality (5.2) follows from (5.4) by applying a standard covering
argument in the annulus ≤ |x| ≤ (1 + ϑ).
5.1. Universal estimates
The following lemma is due to Mitidieri and Pohozaev [17].
Lemma 5.4. Let q 1. Assume that u is a weak solution of the problem
∆2
u(x) ≥ |u(x)|q
, x ∈ Ω ⊂ RN
, (5.5)](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-35-320.jpg)
![Harnack Inequality and Biharmonic Equations 25
where C and C1 are positive constants which depend on N. From Lemma 5.4 and
Hölder inequality, we get that
|u(x)| ≤ C1R4−N
0 R
N(q−1)−4q
q−1
0 + C2R−N
0 R
N(q−1)−4q
q(q−1)
0 R
N
q
0 , (5.11)
which implies that for any x ∈ BR(x0)
|u(x)| ≤ C R− 4
q−1 .
Thus (5.9) follows by choosing R = d(x, ∂Ω). A similar argument as above can be
used to show that indeed (5.10) holds. We omit the details.
Acknowledgment
We thank the referee for useful comments.
References
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[5] G. Caristi, E. Mitidieri and R. Soranzo, Isolated Singularities of Polyharmonic Equa-
tions, Atti del Seminario Matematico e Fisico dell’Università di Modena, Suppl. al
Vol. XLVI (1998), pp. 257–294.
[6] Z.-Q. Chen, Z. Zhao, Harnack principle for weakly coupled elliptic systems, J. Diff.
Eq. 139 (1997), 261–282.
[7] F. Chiarenza, E. Fabes, N. Garofalo, Harnack’s inequality for Schrödinger operators
and the continuity of solutions, Proc. Amer. Math. Soc. 98 (1986), 415–425.
[8] E.B. Fabes and D.W. Stroock, The Lp
-integrability of Green’s function and fun-
damental solutions for elliptic and parabolic equations, Duke Math. J., 51 (1984),
997–1016.
[9] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic
Systems, Annals of Mathematics Studies, Princeton University Press, 1983.
[10] B. Gidas, J. Spruck, Global and Local behavior of Positive Solutions of Nonlinear
Elliptic Equations, Comm. Pure Appl. Math., 34 (1981), 525–598.
[11] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order,
Springer Verlag, 1983.
[12] H.-Ch. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators
with Dirichlet boundary conditions, Math.Ann. 307 (1997), 589–626.](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-37-320.jpg)
![26 G. Caristi and E. Mitidieri
[13] A.M. Hinz, H. Kalf, Subsolution estimates and Harnack inequality for Schrödinger
operators, J. Reine Angew. Math., 404 (1990), 118–134.
[14] H. Mâagli, F. Toumi, M. Zribi, Existence of positive solutions for some polyhar-
monic nonlinear boundary-value problems, Electronic Journal of Differential Equa-
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[15] F. Mandras, Diseguaglianza di Harnack per i sistemi ellittici debolmente accoppiati,
Bollettino U.M.I., 5 14-A (1977), 313–321.
[16] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in RN
,
Differential and Integral Equations, Vol. 9, N. 3, (1996), 465–479.
[17] E. Mitidieri, S.I. Pohozaev, A priori estimates and nonexistence of solutions to non-
linear partial differential equations and inequalities, Tr. Math.Inst. Steklova, Ross.
Akad. Nauk, vol. 234, 2001.
[18] M. Nicolescu, Les Fonctions Polyharmoniques, Hermann and Cie Editeurs, Paris
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tors and related topics, Math. Z. 203 (1990), 129–152.
[24] Ch.G. Simader, Mean value formula, Weyl’s lemma and Liouville theorems for ∆2
and Stoke’s System, Results Math. 22 (1992), 761–780.
[25] R. Soranzo, Isolated Singularities of Positive Solutions of a Superlinear Biharmonic
Equation, Potential Analysis, 6 (1997), 57–85.
[26] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second
ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189–258.
[27] N.S. Trudinger, On Harnack type inequalities and their application to quasilinear
elliptic partial differential equations, Comm. Pure Appl. Math., 20 (1967), 721–747.
[28] Z. Zhao, Conditional gauge with unbounded potential, Z. Wahrsch. Verw. Gebiete, 65
(1983), 13–18.
[29] Z. Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac
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mun. Math. Phys., 93 (1984), 19–31.
Gabriella Caristi and Enzo Mitidieri
Dipartimento di Matematica e Informatica
Università di Trieste
I-34127 Trieste, Italy
e-mail: caristi@univ.trieste.it
e-mail: mitidier@univ.trieste.it](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-38-320.jpg)
![28 C. Carstensen
Indeed, many second-order elliptic boundary value problems are recast in a
weak form
a(u, v) = b(v) for all v ∈ V,
where (V, a) is some Hilbert space with induced norm · a and b ∈ V ∗
has the
Riesz representation u ∈ V . Given a subspace Vh, the discrete solution uh ∈ Vh
satisfies
a(uh, vh) = b(vh) for all vh ∈ Vh.
The error e := u − uh ∈ V then satisfies the best-approximation property
e a = min
vh∈Vh
u − vh a.
(The proof is given at the end of Subsection 3.1 below for completeness.)
The classical estimation of the upper bound u − vh a replaces vh by the
nodal interpolation operator applied to the exact solution u. In the applications,
one typically has
u ∈ V ∩ Hs
(Ω; Rm
)
for some Sobolev space Hs
(Ω; Rm
) on some domain Ω and some regularity pa-
rameter s ≥ 1. The higher regularity with s 1 is subject to regularity theory of
elliptic PDEs and s depends on the smoothness of data and coefficients as well as
on the boundary conditions and on the geometry (e.g., corners) of Ω. An optimal
regularity for 2D and convex domains or C2
boundaries is s = 2. For smaller s
and/or for higher space dimensions, there holds
u ∈ V ∩ Hs
(Ω; Rm
) ⊂ C(Ω; Rm
).
Thus the nodal interpolation operator, which evaluates the function u at a finite
number of points, is not defined well.
The Clément operators J(v) cures that difficulty in that they replace the
evaluation at discrete points by some initial local best approximation of v in the
L2
sense followed by the point evaluation of this local best approximation at the
discrete degrees of freedom.
This small modification is essential in order to receive approximation and
stability properties in the sense that
v − J(v) Hr (Ω) ≤ C hs−r
v Hs(Ω)
with some mesh-size independent constant C 0 and the maximal mesh-size
h 0. The point is that the range for the exponents s and r is possible with
r = 0, 1 and s = 1, 2 while s = 1 is excluded for the nodal interpolation operator
even for dimension n ≥ 2.
At the time of Clément’s research, this improvement seemed to be regarded
as a marginal technicality in the a priori error control – this possible underestima-
tion is displayed in Ciarlet’s finite element book [17] where Clément’s research is
summarized as an exercise (3.2.3).](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-40-320.jpg)
![Clément Interpolation in AFEM 29
The relevance of Clément’s results was highlighted in the a posteriori er-
ror analysis of the last two decades where the aforementioned first-order ap-
proximation and stability property of J for r ≤ s = 1 had been exploited.
Amongst the most influential pioneering publications on a posteriori error con-
trol are [2, 21, 5, 19] followed by many others. The readers may find it rewarding
to study the survey articles of [20, 7] and the books of [26, 1, 3, 4] for a first insight
and further references.
This paper gives a very general approach to Clément-type interpolation in
Section 2 and comments on its applications in the a posteriori error analysis in
Section 3 where further applications such as the error reduction in adaptive finite
element schemes and the coarsening are seen as currently open fields.
2. Clément-type approximation
This section aims at a first glance of a class of approximation operators J :
W1,p
(Ω) → W1,p
(Ω) with discrete image which are named after Philippe Clément
and called Clément-type interpolation operators or sometimes quasi-interpolation
operators.
The most relevant properties include the local first order approximation, i.e.,
for all v ∈ V ⊂ W1,p
(Ω) with weak gradient Dv ∈ Lp
(Ω)n
in Ω ⊂ Rn
and an
underlying triangulation with local mesh-size h ∈ L∞
(Ω) and discrete space Vh,
there holds
v − J(v) Lp(Ω) ≤ C hDv Lp(Ω) and h−1
(v − J(v)) Lp(Ω) ≤ C Dv Lp(Ω),
and the W1,p
(Ω) stability property, i.e., for v ∈ V ,
DJ(v) Lp(Ω) ≤ C Dv Lp(Ω).
A discussion on a particular operator with an additional orthogonality property
ends this section. Although the emphasis in this section is not on the evaluation
of constants and their sharp estimation, there is a list M1, . . . , M5 of highlighted
relevant parameters.
Definition 2.1 (abstract assumptions). Let Ω be a bounded Lipschitz domain in
Rn
, let 1 p, q ∞ with 1/p + 1/q = 1 and m, n ∈ N = {1, 2, 3, . . .}. Suppose
that (ϕz : z ∈ N) is a Lipschitz continuous partition of unity [associated to Vh
below] on Ω with
ωz := {x ∈ Ω : ϕ(x) = 0} ⊆ Ωz ⊂ Ω.
The sets Ωz are supposed to be open, nonvoid, and connected supersets of ωz of
volume |Ωz|. For any z ∈ N let Az ⊆ Rm
be nonvoid, convex, and closed and let
Πz : Rm
→ Rm
be the orthogonal projection onto Az in Rm
(with respect to the
Euclidean metric). Let
V ⊆ W1,p
(Ω; Rm
) and Vh := {
z∈N
azϕz : az ∈ Az}.](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-41-320.jpg)
![Clément Interpolation in AFEM 31
There holds (H1) with Poincaré and Friedrich’s inequalities and
Hz :=
cP (ωz) if z ∈ N ∩ Ω,
cF (ωz, (∂Ω) ∩ (∂ωz)) if z ∈ N ∩ ∂Ω;
Hz is of the form global constant times diam(ωz). It is always true that
M1 = M2 = 1 = M4 and M3 = 3.
Since Dϕz L∞(Ω) ≈ H−1
z there holds
M5 1.
Notice that M5 may be very large for small angles in the triangulation while the
constants M1, . . . , M4 are robust with respect to small aspect ratios.
The aforementioned example is not exactly the choice of [18] but certainly
amongst the natural choices [14, 25].
The announced approximation and stability properties are provided by the
following main theorem.
Theorem 2.4. The map J : V → Vh; v →
z∈N Πz(Jz(v|Ωz ))ϕz satisfies
(a) ∃c1 0 ∀v ∈ V, v − J(v) Lp(Ω) ≤ c1 HDv Lp(Ω);
(b) ∃c2 0 ∀v ∈ V, H−1
(v − J(v)) Lp(Ω) ≤ c2 Dv Lp(Ω),
(c) ∃c3 0 ∀v ∈ V, DJ(v) Lp(Ω) ≤ c3 Dv Lp(Ω).
The constants c1 and c2 depend on M1, . . . , M4 and on p, q, m, n while c3 depends
also on M5.
Proof. Given v ∈ V , let vz := Jz(v|Ωz ), z ∈ N, be the coefficient in
J(v) =
z∈N
Πz(vz)ϕz
and let vz := |Ωz|−1
Ωz
v(x) dx be the local integral mean.
The first step of this proof establishes the estimate
v − Πz(vz) Lp(Ωz ) ≤ M4Hz Dv Lp(Ωz ). (2.1)
The assumptions (H1) and (H2) lead to
v|Ωz − Πz(vz) Lp(Ωz) ≤ Hz Dv Lp(Ωz ),
vz − vz Lp(Ωz) ≤ ezHz Dv Lp(Ωz ).
Since Πz is non-expansive (Lipschitz with Lip(Πz) ≤ 1; Rm
is endowed with the
Euclidean norm used in Lp
(Ω; Rm
)) there holds
Πz(vz) − Πz(vz) Lp(Ωz ) ≤ vz − vz Lp(Ω).
The combination of the aforementioned estimates yields an upper bound of the
right-hand side in
v − Πz(vz) Lp(Ωz) ≤ v − Πz(vz) Lp(Ωz ) + Πz(vz) − Π(vz) Lp(Ωz )
and, in this way, establishes (2.1) and concludes the first step.](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-43-320.jpg)
![Clément Interpolation in AFEM 37
In fact, given any v ∈ V with v a = 1, the identity (3.2) follows from
1 − a(e/ e a, v) = 1
2 a(e/ e a, e/ e a) − a(e/ e a, v) + 1
2 a(v, v)
= 1
2 v − e/ e a
2
a.
The interpretation of the error estimate (3.2) is as follows. The maximizing v
in (3.1) (i.e., v ∈ V with maximal R(v) subject to v a ≤ 1) is unique and
equals e/ e a. As a consequence, the computation of the maximizing v in (3.1)
is equivalent to and indeed equally expensive as the computation of the unknown
e/ e a and so essentially of the exact solution u.
Therefore, a posteriori error analysis aims to compute lower and upper bounds
of R V ∗ instead of its exact value. However, the use of any good guess of the
exact solution from some postprocessing with extrapolation or superconvergence
phenomenon is welcome (and, e.g., may enter v).
3.3. Residual representation formula
Without stating examples in explicit form, it is stressed that linear and nonlinear
boundary value problems of second order elliptic PDEs typically lead (after some
partial integration) to some explicit representation of the residual R := b−a(uh, ·).
Let T denote the underlying regular triangulation of the domain and let E denote
a set of edges for n = 2. Then, the data and the discrete solution uh lead to given
quantities rT ∈ Rm
, the explicit volume residual, and rE ∈ Rm
, the jump residual,
for any element domain T and any edge E such that the residual representation
formula
R(v) =
T ∈T
T
rT · v dx −
E∈E
E
rE · v ds (3.3)
holds for all v ∈ V .
The goal of a posteriori error control is to establish guaranteed lower and
upper bounds of the (energy norm of the) error and hence of the (dual norm of
the) residual.
Any choice of v ∈ V {0} immediately leads to a lower error bound via
η := |R(v)/ v a| ≤ e a.
Indeed, edge and element-oriented bubble functions are studied for v in [26] to
prove efficiency.
More challenging appears reliability, i.e., the design of upper error bounds
which require extra conditions. In fact, the Galerkin orthogonality Vh ⊂ kern R is
supposed to hold for the data in (3.3).
3.4. Explicit residual-based error estimators
Given the explicit volume and jump residuals rT and rE in (3.3), one defines the
explicit residual-based estimator
η2
R :=
T ∈T
h2
T rT
2
L2(T ) +
E∈EΩ
hE rE
2
L2(E), (3.4)](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-49-320.jpg)
![40 C. Carstensen
Notice for f ∈ H1
(Ω)m
and the mesh-size hT
∈ P0(T ) there holds
osc(f) ≤ C h2
T
Df L2(Ω)
and so osc(f) is of quadratic and hence of higher order. We refer to [10, 9, 16], [22],
[6], and [24] for further details on and proofs of (3.8). The proof in [16] is based on
Theorem 2.6. In fact, the arguments of Subsection 3.4 are combined with rT = f
for the lowest-order finite element method at hand. Then, Theorem 2.6 shows
Ω
fw dx ≤ c4 Dv L2(Ω) osc(f)
and leads to the announced higher-order term.
Dominating edge contributions lead to efficiency of averaging schemes [10, 11].
3.6.2. Reliability constants. The global constants in the approximation and sta-
bility properties of Clément-type operators enter the reliability constants. The
estimates of [14] illustrate some numbers as local eigenvalue problems. Based on
numerical calculations, there is proof that, for all meshes with right-isosceles tri-
angles and the Laplace equation with right-hand side f ∈ L2
(Ω), there holds
reliability in the sense of
D(u − uh) L2(Ω) ≤
T ∈T
h2
T f 2
L2(T )
1/2
+
E∈E
hE
E
[∂uh/∂νE]2
ds
1/2
with an undisplayed constant Crel 1 in the upper bound [15].
The aspect of upper bounds with constants and corresponding overestimation
is empirically discussed in [13].
3.6.3. Coarsening. One actual challenging detail is coarsening. Reason number
one is that coarsening cannot be avoided for time evolving problems to capture
moving singularities. Reason number two is the design of algorithms which are
guaranteed to convergence in optimal complexity [8].
Any coarsening excludes the Pythagoras theorem (which assumes V ⊂ V +1)
and so perturbations need to be controlled which leads to the estimation of
min
v+1∈V+1
v +1 − u a.
Ongoing research investigates the effective estimation of this term.
3.6.4. Other norms and goal functionals. The previous sections concern estima-
tions of the error in the energy norm. Other norms are certainly of some interest
as well as the error with respect to a certain goal functional. The later is some
given bounded and linear functional : V → R with respect to which one aims to
monitor the error. That is, one wants to find computable lower and upper bounds
for the (unknown) quantity
|(u) − (uh)| = |(e)|.](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-52-320.jpg)
![Clément Interpolation in AFEM 41
Typical examples of goal functionals are described by L2
functions, e.g.,
(v) =
Ω
v dx ∀v ∈ V
for a given ∈ L2
(Ω) or as contour integrals. To bound or approximate J(e) one
considers the dual problem
a(v, z) = (v) ∀v ∈ V (3.9)
with exact solution z ∈ V (guaranteed by the Lax-Milgram lemma) and the dis-
crete solution zh ∈ Vh of
a(vh, zh) = (vh) ∀vh ∈ Vh.
Set f := z − zh. Based on the Galerkin orthogonality a(e, zh) = 0 one infers
(e) = a(e, z) = a(e, z − zh) = a(e, f). (3.10)
Cauchy inequalities lead to the a posteriori estimate
|(e)| ≤ e a f a ≤ ηuηz.
Indeed, utilizing the primal and dual residual Ru and Rz in V ∗
, defined by
Ru := b − a(uh, ·) and Rz := − a(·, zh),
computable upper error bounds for e V ≤ ηu and f V ≤ ηz can be found by
the arguments of the energy error estimators [1, 3]. Indeed, the parallelogram rule
shows
2 (e) = 2 a(e, f) = e + f 2
a − e 2
a − f 2
a.
This right-hand side can be written in terms of residuals, in the spirit of (3.1),
namely e a = Ru V ∗ , f a = Rz V ∗ , and
e + f a = Ru+z V ∗
for
Ru+z := b + − a(uh + zh, ·) = Ru + Rz ∈ V ∗
.
Therefore, the estimation of (e) is reduced to the computation of lower and upper
error bounds for the three residuals Ru, Rz, and Ru+z with respect to the energy
norm. This illustrates that the energy error estimation techniques of the previous
sections may be employed for goal-oriented error control [1, 3].
Alternatively, Rannacher et al. investigated the weighted residual technique
where the Clément interpolation operator is employed as well. The reader is re-
ferred to [4].](https://image.slidesharecdn.com/442418-250518143527-4e14ccf6/85/Partial-Diff-Equations-And-Functional-Analysis-The-Philippe-Clement-Festschrift-Koelink-E-53-320.jpg)
![42 C. Carstensen
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1587.
[17] Ciarlet, P.G. The Finite Element Method for Elliptic Problems. North-Holland, 1978.
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Partial Diff Equations And Functional Analysis The Philippe Clement Festschrift Koelink E
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- 8. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Portrait of Philippe Clément . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Philippe Clément: Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Gabriella Caristi and Enzo Mitidieri Harnack Inequality and Applications to Solutions of Biharmonic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Carsten Carstensen Clément Interpolation and Its Role in Adaptive Finite Element Error Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Sandra Cerrai Ergodic Properties of Reaction-diffusion Equations Perturbed by a Degenerate Multiplicative Noise . . . . . . . . . . . . . . . . . . . . . . 45 Giuseppe Da Prato and Alessandra Lunardi Kolmogorov Operators of Hamiltonian Systems Perturbed by Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A.F.M. ter Elst, Derek W. Robinson, Adam Sikora and Yueping Zhu Dirichlet Forms and Degenerate Elliptic Operators . . . . . . . . . . . . . . . . . . . 73 Onno van Gaans On R-boundedness of Unions of Sets of Operators . . . . . . . . . . . . . . . . . . . 97 Matthias Geißert, Horst Heck and Matthias Hieber On the Equation div u = g and Bogovskiı̆’s Operator in Sobolev Spaces of Negative Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 F. den Hollander Renormalization of Interacting Diffusions: A Program and Four Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Tuomas P. Hytönen Reduced Mihlin-Lizorkin Multiplier Theorem in Vector-valued Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
- 9. vi Contents Stig-Olof Londen Interpolation Spaces for Initial Values of Abstract Fractional Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Noboru Okazawa Semilinear Elliptic Problems Associated with the Complex Ginzburg-Landau Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Jan Prüss and Gieri Simonett Operator-valued Symbols for Elliptic and Parabolic Problems on Wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Jan Prüss and Mathias Wilke Maximal Lp-regularity and Long-time Behavior of the Non-isothermal Cahn-Hilliard Equation with Dynamic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Jacques Rappaz Numerical Approximation of PDEs and Clément’s Interpolation . . . . . . 237 Erik G.F. Thomas On Prohorov’s Criterion for Projective Limits . . . . . . . . . . . . . . . . . . . . . . . . 251 Lutz Weis The H∞ Holomorphic Functional Calculus for Sectorial Operators – a Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
- 10. Preface The present volume is dedicated to Philippe Clément on the occasion of his re- tirement in December 2004. It has its origin in the workshop “Partial Differen- tial Equations and Functional Analysis” (Delft, November 29–December 1, 2004) which was held to celebrate Philippe’s profound contributions in various areas of Mathematical Analysis. The articles presented here offer a panorama of current developments in the theory of partial differential equations as well as applications to such diverse ar- eas as numerical analysis of PDEs, Volterra equations, evolution equations, H∞ - calculus, elliptic systems, mathematical physics, and stochastic analysis. They re- flect Philippe’s interests very well and indeed several of the authors have collabo- rated with him in the course of his career. The editors gratefully acknowledge the financial support of the Royal Nether- lands Academy of Arts and Sciences, the Netherlands Organization for Scientific Research, and the Thomas Stieltjes Institute for organizing the workshop. They also thank Thomas Hempfling for the pleasant collaboration during the prepara- tion of this volume. Last but not least the editors, all members of his former group, thank Philippe for his constant inspiration and for sharing his enthusiasm in mathematics with them. The Editors
- 11. This volume is dedicated to Philippe Clément on the occasion of his retirement.
- 12. Philippe Clément: Curriculum Vitae Philippe Clément, born on 9 January 1943 in Billens, Switzerland, started his study in Physics at the Ecole Polytechnique de l’ Université de Lausanne (now EPFL) in 1962 and obtained the degree of Physicist-Engineer in 1967. During that period he discovered that his true interest was much more in Mathematics and he obtained the License des Sciences Mathématiques in 1968 from the University of Lausanne. Hereafter he started to work on his Ph.D. thesis in the area of Numerical Analysis at the EPFL with J. Descloux as supervisor. He defended his thesis, “Méthode des éléments finis appliquée à des problèmes variationnels de type indéfini”, in Febru- ary 1974. Some of the results were published in his seminal paper “Approximation by finite element functions using local regularization” (Rev. Française Automat. Informat. Recherche Opérationelle, RAIRO Analyse Numérique 9, 1975, R-2, 77– 84). In this paper he introduced what is nowadays known in the literature as the Clément-type interpolation operators, which play a key role in the analysis of adaptive finite element methods. In the period 1972–74 Philippe was First Assistant at the Department of Mathematics of the EPFL and under the influence of B. Zwahlen he became in- terested in Nonlinear Analysis. It was a very stimulating and inspiring time and environment for him, in particular, he met at various workshops Amann, Aubin, Da Prato, Grisvard, Tartar and others. The years 1974–77 Philippe continued his mathematical work, supported by the Swiss National Foundation for Scientific Research, in Madison (USA), first as Honorary Fellow at the Mathematics De- partment, later as a Research Staff Member at the Mathematics Research Center, of the University of Wisconsin. In that period he came into contact with Crandall and Rabinowitz and worked on nonlinear elliptic problems. Together with Nohel and Londen, he started to be involved in nonlinear Volterra equations. In 1977 Philippe moved to the University of Technology in Delft, were he was appointed as Associate Professor. In 1980 he became full professor and in 1985 he obtained the Chair in Functional Analysis in Delft. His main areas of interest and research were (and still are) the theory of evolution equations, operator semigroups as well as the Volterra equations and elliptic problems mentioned before. In partic- ular, he was involved in problems concerning maximal regularity and problems re- lated to functional calculus. Philippe is widely recognized for his important contri- butions in these areas. The very stimulating seminars in Delft on the theory of semi- groups have resulted in the book “One-Parameter Semigroups” (Clément, Heij- mans et al.). In recent years his interests also include stochastic integral equations.
- 13. Operator Theory: Advances and Applications, Vol. 168, 1–26 c 2006 Birkhäuser Verlag Basel/Switzerland Harnack Inequality and Applications to Solutions of Biharmonic Equations Gabriella Caristi and Enzo Mitidieri This paper is dedicated to our friend Philippe Clément for “not killing birds” Abstract. We prove Harnack type inequalities for linear biharmonic equa- tions containing a Kato potential. Various applications to local boundedness, Hölder continuity and universal estimates of solutions for biharmonic equa- tions are presented. 1. Introduction During the last decade considerable attention has been paid to solutions of the time-independent Schrödinger equation −∆u = V (x)u, x ∈ Ω ⊆ RN , (1.1) where Ω is an open subset of RN and the potential V belongs to the Kato class KN,1 loc (Ω). See, e.g., Aizenman and Simon [1], Zhao [28], [29], [30], Fabes and Strook [8], Chiarenza, Fabes and Garofalo [7], Hinz and Kalf [13], Serrin and Zou [22], Simader [23] and the references therein. Following different approaches these au- thors have studied the regularity of the solutions and proved a Harnack-type inequality. Such inequality in turn can be used to prove results such as strong maximum principles, removable point singularities, existence of solutions for the Dirichlet problem, Liouville theorems and universal estimates on solutions for non- linear equations. In the present paper we shall discuss weak solutions of the following fourth- order elliptic equation ∆2 u = V u, x ∈ Ω ⊆ RN , (1.2) where the potential V is nonnegative in Ω and belongs to KN,2 loc (Ω) the natural Kato class of potentials associated to the biharmonic operator. See Definition 2.2 in the next section. We point out that Kato classes of potentials associated to polyharmonic operators and some generalizations have been used in a series
- 14. 2 G. Caristi and E. Mitidieri of works by Bachar et al. see [3], [4] and Maagli et al. [14]. In these papers the authors prove various interesting results including 3G type Theorems and existence of positive solutions for second order and semilinear polyharmonic equations. Here we are interested in several kinds of results of qualitative nature and mainly some of the consequences that can be deduced from Harnack inequality that we are going to prove during the course. The first one is the regularity problem which includes the local boundedness and the Hölder continuity of solutions. The second kind of results are Harnack-type inequalities. The prototype version of these states: There exist constants C = C(N) and r 0 depending on Ω and norms of V such that all solutions of (1.2) with u ≥ 0, −∆u ≥ 0 in Ω (i.e., in the sense of distributions in Ω) satisfy sup Br/2 u(x) ≤ C inf Br/2 u(x), (1.3) where Br/2 denotes any ball contained in Ω. Our interest in these results was stimulated by studying certain nonlinear biharmonic equations and their isolated singularities (see [5], [25]). For several questions concerning the nature of non-removable singularities and the behavior of a positive solution in a neighborhood of the isolated singularity it is customary to assume that V ∈ Lq loc (Ω) for q N 4 . On the other hand the technique for proving Lp -estimates for all p ∞, relating the sup u to certain integrals involving the solution of (1.2) as employed by Serrin [20], [21], Stampacchia [26], Trudinger [27] for second order elliptic equations and by Mandras [15] in the context of weakly coupled linear elliptic systems, seems not to be easily extendable for this kind of equations. In order to derive a local a priori majorization for the sup u, we shall follow two different approaches. The first one was introduced by Simader in [23] and it is based on representation formulae of solutions and the 3G theorem of Zhao [30], while the latter have been used by Chiarenza et al [7] and its main ingredients are Caccioppoli-type inequalities and maximum principle arguments for the operator ∆2 − V . We remark that the Harnack inequality (1.3) admits a rather simple proof in the special case V ≡ 0, that is for the biharmonic equation ∆2 u = 0 in Ω. In fact, due to the special type of mean value formulas for biharmonic functions (see formula (3.4)0 of Simader [24]) u(x) = N + 1 2 1 |BR(x)| BR(x) u(y) (N + 2) − (N + 3) |y − x| R dy it turns out that for x0 ∈ Ω and B2r(x0) ⊂ Ω sup Br/2(x0) |u(x)| ≤ c(N) |Br(x0)| Br(x0) |u(y)| dy,
- 15. Harnack Inequality and Biharmonic Equations 3 where c(N) = 2N−1 (N + 1) (2N + 5). On the other hand both conditions u ≥ 0 and −∆u ≥ 0 imply (see [11]) inf Br/2(x0) u(x) ≥ (2/3)N |Br(x0)| Br(x0) u(y) dy, hence inequality (1.3) follows immediately. We observe that in general there is no hope of obtaining a Harnack inequality for solutions of (1.2) under the only assumption that they are nonnegative. A simple example in this direction is given by u(x) = x2 1 in Ω = B2(0). It is interesting to note that the sign conditions (−∆)m u ≥ 0 in Ω, for m = 1, . . . , k − 1, can be already found in the classical book of Nicolescu [18], p. 16, in the context of polyharmonic equations of the type ∆k u = 0 in Ω, k ∈ N (see also [2]). This paper is organized as follows. Section 2 contains few preliminary facts, the proof of the representation formula (see Lemma 2.6) (following Simader [23]) for weak solutions of the problem, (−∆)m u = V u, x ∈ Ω ⊆ RN , (1.4) where V ∈ KN,m loc (Ω), N 2m, m ≥ 2, and some remarks on Green functions for Schrödinger biharmonic operators. In Section 3 we prove the results on local boundedness and continuity of solutions of (1.2) and a related Harnack inequality (1.3), (see Theorem 3.6). Namely, for each p ∈ (0, ∞) there exists a constant C = C(p) and r 0 depending on Ω and some local norms of V such that sup Br/2(x0) |u(x)| ≤ C 1 |Br(x0)| Br(x0) |u(y)|p dy 1/p . (1.5) In Section 4 we briefly use the approach by Chiarenza, Fabes and Garofalo [7], adapted to fourth order problems of the type (1.2). The main outcome of this technique is an estimate of the modulus of continuity of the solutions is given in Theorem 4.12 and the Hölder continuity is established in Theorem 4.13 for a class of potentials V which includes those that belongs to the Lebesgue spaces Lp loc (Ω) for p N 4 . Finally Section 5 is devoted to some consequences of Theorem 4.9. The first application is another proof of Theorem 3.6 while Theorem 5.2 concerns the behavior at infinity of solutions u ∈ Lp RN . Also, in Theorem 5.1 and Theorem 5.3 we prove respectively a Harnack inequality of the type (1.3) and a modified version of it in the limiting case V = O 1 |x|4 and Ω = BR(0){0}. We conclude the paper with an application to universal estimates for a semi- linear biharmonic equation in a general domain (see Theorem 5.5). This result can be considered a step towards to the general understanding of the existence of universal estimates for solutions of elliptic systems containing non-linearities with growth below the so called first critical hyperbola (see [16], [17]).
- 16. 4 G. Caristi and E. Mitidieri 2. Notation and preliminaries In the sequel, Ω will denote a nonempty open subset of RN . If Ω1 ⊂ Ω2 ⊆ RN are open sets, we write Ω1 ⊂⊂ Ω2 if and only if Ω1 is compact and Ω1 ⊂ Ω2. For x ∈ RN and r 0 we write Br(x) to denote the open ball of center x and radius r and ∂Br(x) to denote its boundary. In what follows we assume that j is a given nonnegative function in C∞ 0 (RN ) such that j(z) = 0 if |z| ≥ 1 and: RN j(x)dx = 1. For any 0 we define j(x) = −N j(−1 x). Definition 2.1. Given any function f ∈ Lq (Ω) with 1 ≤ q ∞, for any 0 we define the mollified function by f(x) = RN j(x − y)f(y)dy. From the definition it follows that f ∈ C∞ (Ω)∪Lq (Ω) and f−f Lq(Ω) → 0, as → 0. Definition 2.2. Given N 2m, the Kato class KN,m loc (Ω) is the set of functions V ∈ L1 loc(Ω) such that for any compact set K ⊂ Ω the quantity φV (t, K) = sup x∈RN Bt(x) |V (y)|χK (y) |x − y|N−2m dy is finite (here, χK denotes the characteristic function of K) and lim t→0 φV (t, K) = 0. Example 2.3. Any function satisfying a local Stummel condition. We recall that V satisfies a local Stummel-condition if it is measurable in Ω and there exists γ ∈ (0, 4) such that for each ω ⊂⊂ Ω there exists a constant Dω 0 such that sup x∈RN ω∩B1(x) |V (y)| 2 |x − y|N−4m+γ dy ≤ Dω. We claim that if this condition is satisfied, then, V ∈ KN,m loc (Ω). Indeed, for BR(x0) ⊂⊂ Ω and 0 t ≤ 1 we have Bt(x)∩BR(x0) |V (y)| |x − y| N−2m dy ≤ Bt(x)∩BR(x0) |V (y)| 2 |x − y|N−4m+γ dy 1 2 Bt(x) dy |x − y|N−γ 1 2 ≤ D 1 2 BR(x0) σN Nγ 1 2 t γ 2 . Therefore Definition 2.2 is satisfied with φV (t) = C(D, γ) tγ/2 .
- 17. Harnack Inequality and Biharmonic Equations 5 Example 2.4. Any V belonging to Lα loc(Ω) with α N/2m. In fact, in this case if Bt(x) ⊂⊂ Ω we have: Bt(x) V (y) |x − y|N−2m dy ≤ Bt(x) |V (y)|α dy 1 α Bt(x) 1 |x − y|(N−2m)α dy 1 α ≤ σ 1 α N V Lα(Bt(x))t2m− N α . (2.1) Therefore, Definition 2.2 is satisfied with φV (t) = C(V ) t2m− N α for t 0. Definition 2.5. We say that u ∈ L1 loc(Ω) is a distributional solution of (1.4), or that u is a solution in the sense of distributions, if u ∈ L1 loc(Ω) and for any ψ ∈ C∞ 0 (Ω) we have Ω u(y)∆m ψ(y)dy = Ω V (y)u(y)ψ(y)dy (2.2) For any m ≥ 1, N 2m and r 0 we set Φm,N (r) = Cm,N r2m−N , (2.3) where Cm,N = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Γ(2 − N/2) 22m−2(m − 1)!Γ(m + 1 − N/2) , if N is odd, (−1)m−1 (N/2 − m − 1)! 22m−2(m − 1)!(N/2 − 2)! , if N is even. The function Φm,N as function of r = |x|, for x ∈ RN {0}, is polyharmonic of degree m and is called the fundamental solution of the equation ∆m u = 0. Here, the constants Cm,N are chosen so that ∆p Φm,N = Φm−p,N for p = 1, . . . , m − 1, (see [2]). In the sequel, sometimes we will write Φm(x, y) instead of Φm,N (|y − x|). The following representation formula holds: Lemma 2.6. Let N 2m. If u ∈ C∞ 0 (RN ), then for any x ∈ supp(u) u(x) = −ΩN RN Φm,N (|y − x|)∆m u(y)dy, (2.4) where ΩN = ((N − 2)ωN )−1 and ωN = 2πN/2 /Γ(N/2) is the area of the unit sphere.
- 18. 6 G. Caristi and E. Mitidieri Proof. In order to prove (2.4) we apply the second Green identity in Dρ = RN Bρ(y) successively, add the results and obtain u(x) = ΩN m−1 l=0 ∂Dρ ∂Φl+1(x, y) ∂νy ∆l u(y) − ∂∆l u(y) ∂ν Φl+1(x, y) dy −ΩN Dρ Φm(x, y)∆m u(y)dy. (2.5) By a standard argument it can be proved that all the nonzero boundary terms tend to 0 as ρ → 0. Throughout the paper, for the sake of clarity, we will assume that m = 2 and N 4, that is, we shall consider the problem ∆2 u = V (x)u, x ∈ Ω ⊆ RN , N 4. (2.6) The proofs of the extensions of the results to the general case can be obtained by obvious modifications. Lemma 2.7. Let g ∈ L1 loc(RN ) and u ∈ L1 loc(RN ) be a distributional solution of ∆2 u = g. Then, for any ρ ∈ C∞ 0 (Ω) the following formula holds: cN u(x)ρ(x) = − Φ2(x, y)g(y)ρ(y)dy +2 u(y) (∇∆ρ(y), ∇Φ2(x, y)) dy + 2 ∆2 ρ(y) u(y)Φ2(x, y)dy +2 ∆ (∇Φ2(x, y), ∇ρ(y)) u(y)dy + (∇ρ(y), ∇∆Φ2(x, y)) u(y)dy +2 u(y)∆Φ2(x, y)∆ρ(y)dy − u(y)∆2 ρ(y)Φ2(x, y)dy, (2.7) where cN = Ω−1 N . Proof. For any 0, let u be the mollified function of u. Then, given ρ ∈ C∞ 0 (Ω) we can apply Lemma 2.6 and get cN u(x)ρ(x) = − Ω Φ2(x, y)∆2 (uρ)(y)dy. (2.8) Now, we have ∆2 (uρ) = ∆2 (u)ρ + 2(∇∆ρ, ∇u) + 2∆ρ∆u + 2(∇∆u, ∇ρ) + 2∆(∇ρ, ∇u) + u∆2 ρ, (2.9) and we know that ∆2 u = g and that g → g in L1 loc(Ω). To obtain (2.7), first we integrate by parts, taking into account of the fact that all the integrals extended to the boundary of Ω are equal to 0, since supp(ρ) ⊂ Ω. Then, we take the limit as → 0 and apply Lemma 2.8 below.
- 19. Harnack Inequality and Biharmonic Equations 7 Lemma 2.8. Let Ω ⊆ RN be bounded, ρ ∈ C0 0 (Ω), g ∈ L1 loc(Ω) and 0 α N. Then, if for any n ∈ N, n 0 and n → 0, as n → ∞, there exists a subsequence, which we still denote by n, such that Ω gn (y)ρ(y)|y − x|−α dy → Ω g(y)ρ(y)|y − x|−α dy, a.e. in RN . We refer to [23] for its proof. 2.1. Green functions of Schrödinger biharmonic operators We recall some properties of the Green functions associated to the following bound- ary value problems for the biharmonic equation on the ball Br(0) ⊂ RN : ∆2 u(x) = f(x), x ∈ Br(0) : (2.10) that is, the Navier boundary value problem (N): u = ∆u = 0 on ∂Br(0), (2.11) and the Dirichlet boundary value problem (D): u = ∂u ∂ν = 0 on ∂Br(0). (2.12) It is well known that both problems admit a nonnegative Green function on Br(0), which we denote respectively by GN r (x, y) and GD r (x, y). Moreover, the following estimates hold: Lemma 2.9 (3G-Lemma). Let G = GN r or GD r . Then, there exists a constant C1 0 depending only on the dimension N such that for any ball B of RN we have G(x, z)G(z, y) G(x, y) ≤ C1 |x − z|4−N + |z − y|4−N , for all x, y, z ∈ B. The proof of this lemma for the Dirichlet problem is contained in [12], while for the Navier problem it can be straightforwardly obtained by iteration of the 3G-Lemma for the Laplace operator, see [30]. Now, consider instead of (2.10) the Schrödinger biharmonic equation: ∆2 u(x) = V (x)u(x) + f(x), for x ∈ Br(0), (2.13) where V belongs to the natural Kato class KN,2 loc (Br(0)) associated to ∆2 . The following result extends Lemma 2.3 of [6] to the biharmonic case. Proposition 2.10. Assume that V ∈ KN,2 loc (Br(0)). Then, there exists r1 0 such that if 0 r r1, the problems (2.13)–(2.11) and (2.13)–(2.12) admit a nonneg- ative Green function on Br(0).
- 20. 8 G. Caristi and E. Mitidieri Proof. Let us consider the Navier boundary value problem: the proof in the other case is similar. Set A0(x, y) = GN r (x, y) and define for n ≥ 1 An(x, y) = Br GN r (x, z)V (z)An−1(z, y)dz. We prove by induction that there exists r1 0 such that if 0 r r1 we have |An(x, y)| ≤ 1 3n A0(x, y), n ≥ 0. (2.14) If n = 0, (2.14) holds by definition. Assume that for n 0 (2.14) is true, then we have: |An+1(x, y)| ≤ 1 3n Br GN r (x, z)V (z)GN r (z, y)dz. By assumption V ∈ KN,2 loc (Br(0)) and then, we can apply Lemma 2.9 and obtain that |An+1(x, y)| ≤ 1 3n C1GN r (x, y) Br V (z) |x − z|4−N + |z − y|4−N dz. (2.15) Since V ∈ KN,2 loc , given 0 there exists r1 0 such that if 0 r r1, then Br(y) |V (y)| |x − y|N−4 dy . Using this fact in (2.15), we get that |An+1(x, y)| ≤ 1 3n 2C1GN r (x, y), for any r r1. If we choose = 1/(6C1) we get (2.14). This inequality implies that the series A(x, y) = ∞ n=0 An(x, y) is convergent and that its sum A(x, y) satisfies for all x, y ∈ Br A(x, y) = GN r (x, y) + Br GN r (x, z)V (z)A(z, y)dz (2.16) and 1 2 GN r (x, y) ≤ A(x, y) ≤ 2 GN r (x, z). (2.17) From the last inequality it follows that A(·, ·) ≥ 0. Moreover, applying ∆2 to both sides of (2.16) for each fixed y ∈ Br, we have for x ∈ Br ∆2 A(x, y) = δy(x) + V (x)A(x, y) where δy is the Dirac function supported at y. Moreover, it is easy to check that the boundary conditions are satisfied. Remark 2.11. In particular, from (2.17) it follows that A(·, ·) is nonnegative on sufficiently small balls without any assumption on the sign or on the norm of V .
- 21. Harnack Inequality and Biharmonic Equations 9 By a similar argument, it can also be proved that if we fix the radius of the ball, for instance, equal to 1, then, there exists a nonnegative Green function for problems (2.10)–(2.11) and (2.10)–(2.12) if φV (1, B1) is sufficiently small. 3. Local boundedness and continuity of solutions In this section we will prove the local boundedness and the continuity of distribu- tional solutions of problem (2.6), extending in this way Theorem 2.4 of [23]. The method of proofs is essentially the same. First of all, we choose η ∈ C∞ 0 (R) such that 0 ≤ η(t) ≤ 1, η(t) = 1 if t ∈ [−1/2, 1/2] and η(t) = 0 if |t| 1. Given δ 0, we set ηδ(z) = η(δ−1 |z|), for z ∈ RN . Lemma 3.1. Assume that V ∈ KN,2 loc (Ω). Let x ∈ Ω and 0 δ 1 4 dist(x, ∂Ω). Then, for any x = z Ω |V (y)|ηδ(x − y)ηδ(z − y) |x − y|N−4|y − z|N−4 dy ≤ 2N−3 φV (δ, B3δ(x)) η4δ(x − z) |x − z|N−4 . (3.1) Proof. First of all, we observe that if σ = |x − z| 2δ, then it follows that ηδ(x − y)ηδ(z − y) = 0. Hence, we can assume that |x − z| 2δ. Consequently, σ/2 ≤ δ. Define Ω1 = {y ∈ Ω : |y − x| ≤ σ/2} and Ω2 = {y ∈ Ω : |y − x| ≥ σ/2}, and denote the corresponding integrals by I1 and I2. For y ∈ Ω1 we have that |z − y| ≥ |z − x| − |x − y| ≥ σ/2 and therefore I1 ≤ Ω1 2 σ N−4 |V (y)|ηδ(x − y) |x − y|N−4 dy ≤ 2 σ N−4 φV (δ, B3δ(x)). Similarly, I2 ≤ Ω2 2 σ N−4 |V (y)|ηδ(y − z) |y − z|N−4 dy ≤ 2 σ N−4 φV (δ, B3δ(x)). Since η4δ(|x − z|) = 1 for σ ≤ 2δ, these inequalities imply (3.1). Lemma 3.2. Assume that V ∈ KN,2 loc (Ω) and that u is a solution of (2.6) in the sense of distributions. Let x0 ∈ Ω, R0 0 be such that B2R0 (x0) ⊂⊂ Ω. Choose 0 δ0 R0/4 such that φV (δ) ≡ φV (δ, BR0 (x0)) 22−N cN (3.2) for 0 δ ≤ δ0. Then, for 0 R ≤ R0/2, 0 δ ≤ δ0 and for a.e. z ∈ BR(x0) we have: Ω |V (x)||u(x)|ηδ (x − z) |x − z|N−4 dx ≤ 2N−4 δ4−N V u L1(BR+4δ(x0)) + 2Cδ−N u L1(BR+2δ(x0)), (3.3) where C depends on N and the choice of η.
- 22. 10 G. Caristi and E. Mitidieri Proof. Let x ∈ BR0 (x0) and 0 δ0 R0/4. Since ρ(·) = ηδ(| · −x|) ∈ C∞ 0 (Ω), we can apply the representation formula of Lemma 2.7 and obtain that: cN u(x) = − Φ2(x, y)V (y)u(y)ηδ(y − x)dy + 2 u(y)(∇∆yηδ(y − x), ∇Φ2(x, y))dy + 2 ∆2 yηδ(y − x) u(y)Φ2(x, y)dy + 2 ∆(∇Φ2(x, y), ∇yηδ(y − x))u(y)dy + (∇yηδ(y − x), ∇∆Φ2(x, y))u(y)dy + 2 u(y)∆Φ2(x, y)∆yηδ(y − x)dy − u(y)∆2 ηδ(y − x)Φ2(x, y)dy. (3.4) Taking into account of the definition of ηδ and of Φ2 and denoting by χ the characteristic function of the interval [1/2, 1], we get that for any x ∈ BR(x0) |u(x)| ≤ c−1 N Φ2(x, y)|V (y)||u(y)|ηδ(y − x)dy + Cδ−N |u(y)|χ(δ−1 |y − x|)dy, (3.5) where C = C(N, η) 0. Using (3.5) in the left-hand side of (3.3), we obtain |V (x)||u(x)|ηδ(x − z) |x − z|N−4 dx ≤ c−1 N |V (x)||V (y)||u(y)|ηδ(y − x)ηδ(x − z) |x − z|N−4|x − y|N−4 dxdy + Cδ−N |V (x)|ηδ(x − z) |x − z|N−4 |u(y)|χ(δ−1 |y − x|)dxdy. (3.6) Now, we proceed as in the proof of Lemma 2.3 of [23]. The last integral in (3.6) can be estimated by φV (δ)|u|L1(BR+2δ(x0)). Then, we apply Lemma 2.6 to the first integral and get: c−1 N |V (y)||u(y)| |V (x)|ηδ(y − x)ηδ(x − z) |x − z|N−4|x − y|N−4 dxdy ≤ c−1 N 2N−3 φV (δ, B3δ(x)) |V (y)||u(y)|η4δ(y − z) |y − z|N−4 dydx. (3.7) We remark that φV (δ, B3δ(x)) ≤ φV (δ) and that η4δ(y − z) = ηδ(y − z) + (η4δ(y − z) − ηδ(y − z)),
- 23. Harnack Inequality and Biharmonic Equations 11 hence we get c−1 N |V (y)||u(y)| |V (x)|ηδ(y − x)ηδ(x − z) |x − z|N−4|x − y|N−4 dxdy ≤ c−1 N 2N−3 φV (δ) |V (x)||u(x)|ηδ(x − z) |x − z|N−4 dx +c−1 N 2N−3 φV (δ) |V (y)|(η4δ(y − z) − ηδ(y − z)) |y − z|N−4 |u(y)|dy. (3.8) Since η4δ(y−z)−ηδ(y−z)) = 0 if |y−z| ≤ δ/2 and if |y−z| ≥ 4δ, the last integral in (3.8) can be estimated by δ 2 4−N V u L1(BR+4δ(x0)). (3.9) By assumption, if 0 δ ≤ δ0, then 2c−1 N 2N−3 φV (δ) ≤ 1 and hence, from (3.8) and (3.9) the statement follows. Theorem 3.3. Assume that V ∈ KN,2 loc (Ω) and that u is a solution of (2.6) in the sense of distributions. Let x0 ∈ Ω, R0 0 be such that B2R0 (x0) ⊂⊂ Ω and assume that 0 R1 R0/2 is such that φV (R1, B2R0 (x0)) ≤ (2cN )−1 . Then, for 0 R ≤ R1, 0 δ ≤ δ0 (where δ0 is defined in Lemma 3.2) and x ∈ BR(x0) the following estimate holds |u(x)| ≤ 2N−4 δ4−N V u L1(BR+4δ(x0)) + Cδ−N u L1(BR+2δ(x0)), (3.10) where C depends only on N and the choice of η. Proof. The proof follows directly from the proof of Lemma 2.7. Corollary 3.4. Assume that V ∈ KN,2 loc (Ω) and that u is a solution of (2.6) in the sense of distributions. Then u is locally bounded. The continuity of u follows from its local boundedness. Indeed, in (3.4) all the integrals which include derivatives of ηδ contain no singularity and therefore they define continuous functions. Further, for any 0 δ, the function h(x) = − ΩB(x) Φ2(x, y)V (y)u(y)ηδ(|y − x|)dy is continuous and converges locally uniformly to the first integral in (3.4) as → 0. In order to prove the Harnack inequality, we need the following result which gives an estimate of the local · ∞-norm of u in terms of its · 1-norm.
- 24. 12 G. Caristi and E. Mitidieri Proposition 3.5. Assume that V ∈ KN,2 loc (Ω) and that u is a solution of (2.6) in the sense of distributions. Let x0 ∈ Ω, R0 0 be such that B2R0 (x0) ⊂⊂ Ω. Then, there exists 0 R1 ≤ R0/2 such that for any 0 R R1 the following estimate holds u L∞(BR(x0)) ≤ CR−N u L1(B2R(x0)), (3.11) where C depends only on N. Proof. Let R1 ≤ R0/2 be as in Theorem 2.1. Then, for any 0 R R1: φV (B2R(x0)) ≤ 2−1 cN . (3.12) We shall estimate V u L1(B3R/2(x0)) with the u L1(B2R(x0)). To this aim, we take ρ ∈ C∞ 0 (R) such that 0 ≤ ρ ≤ 1, ρ(·) ≡ 1 in [0, 3R/2] and ρ(t) = 0 if t ≥ 2R. Further, we assume that |ρ(i) (t)| ≤ t−i for i = 1, 4 and any t. From Lemma 2.7 we get that for any x ∈ B2R(x0) cN |u(x)|ρ(x) = |V (y)||u(y)|Φ2(x, y)ρ(|x − y|)dy + |u(y)||R(x, y)|dy, (3.13) where R is defined in terms of the derivatives of ρ and of Φ2. Multiplying (3.13) by |V (x)| and integrating over B ≡ B2R(x0) we get: cN B |V (x)||u(x)|ρ(x)dx ≤ |V (x)| |x − y|N−4 dx |V (y)||u(y)|ρ(y)dy + B 3R 2 ≤|x−y|≤2R |u(y)| |V (x)| |x − y|N−4 dxdy ≤ φV (B2R(x0)) B |V (x)||u(x)|ρ(x)dx + φV (B2R(x0)) B |u(x)|dx. (3.14) From (3.12) and (3.14) we deduce that: B |V (x)||u(x)|ρ(x)dx ≤ C B |u(x)|dx, (3.15) which implies that |V (u)|L1(B 3R 2 (x0)) ≤ c−1 N |u|L1(B2R(x0)). (3.16) Now, we apply Theorem 2.1 with δ = R/8. From (3.10) and (3.16), it follows that for any x ∈ BR(x0): |u(x)| ≤ 2N−4 δ4−N |V (u)|L1(B 3R 2 (x0)) + Cδ−N |u|L1(B2R(x0)) ≤ CRN |u|L1(B2R(x0)), (3.17) where C = C(N).
- 25. Harnack Inequality and Biharmonic Equations 13 3.1. The Harnack inequality Theorem 3.6. Assume that V ∈ KN,2 loc (Ω). Let u be a nonnegative weak solution of (2.6) such that −∆u ≥ 0 in Ω. Then there exist C = C(N) and R0 = R0(V ) such that for each 0 R ≤ R0 and B2R(x0) ⊂⊂ Ω we have sup BR/2(x0) u ≤ C inf BR/2(x0) u, (3.18) where C independent of V and u. Proof. Since u is nonnegative and satisfies −∆u ≥ 0 in Ω, then we have (see, e.g., [11]) inf BR/2(x0) u(x) ≥ C(N) 1 |BR| BR(x0) u(y) dy, for all R 0 such that B2R ⊂ Ω . Then, the conclusion follows from this inequality and (3.11). Remark 3.7. The condition −∆u ≥ 0 in Ω is necessary for the validity of the above Theorem as the following example shows: consider the function u(x) = x2 1. It is nonnegative, satisfies ∆2 u = 0 and ∆u = 2, but (3.18) does not hold for x0 and any R. 4. Local boundedness and continuity of solutions: an alternative approach In this section we shall prove The Harnack inequality following the approach of [7], based on Lp estimates. Assume that m = 2, N 4 and that V ∈ KN,2 loc (Ω). First of all, we recall the following result from [19]: Lemma 4.1. Assume that V ∈ KN,2 loc (Ω). Then, for every 0 there exists a constant C = C(, V ) such that Ω |V | ψ2 ≤ Ω |∆ψ|2 + C Ω |ψ|2 , for all ψ ∈ H2,2 0 (Ω) . (4.1) For later use we need also the following slightly different inequality. If t 0 s ∈ (0, t), then: Bt |V | ψ2 ≤ Bt |∆ψ|2 + C(, V ) (t − s)4 Bt |ψ|2 , for all ψ ∈ H2,2 0 (Bt) . (4.2) When V ∈ L N 4−δ (Ω), δ ∈ (0, 4), a simple proof of the above embedding property (4.2) can be obtained as follows. By the embedding L 2N N−4 (Ω) ∩ L2 (Ω) → L 2N N−4+δ (Ω) and by Sobolev’s inequality ψ L 2N N−4 (Ω) ≤ c(N) ∆ψ L2(Ω) , for all ψ ∈ H2,2 0 (Ω) ,
- 26. 14 G. Caristi and E. Mitidieri it follows that for every 0 Ω |V | ψ2 ≤ (t − s) δ V L N 4−δ (Ω) (t − s) −δ ψ 2 L 2N N−4+δ (Ω) ≤ (t − s) δ V N 4−δ ψ 2 L 2N N−4 (Ω) + C(δ) (t − s) −4 − 4−δ δ ψ 2 L2(Ω) . Hence, by taking 1 = (t − s) δ V L N 4−δ (Ω) we obtain Ω |V | ψ2 ≤ 1 Ω |∆ψ| 2 + C (t − s) 4 Ω |ψ| 2 , (4.3) where C depends on 1, δ and (t − s) δ V N/4−δ. Remark 4.2. As a consequence of (4.1) we deduce that, if u ∈ H2,2 loc (Ω), then, V u ∈ L1 loc (Ω). In fact, let B2r ⊂ Ω and let ϕ ∈ C∞ 0 (B2r) be such that 0 ≤ ϕ ≤ 1, ϕ ≡ 1 on Br. Since uϕ ∈ H2,2 0 (B2r) and B2r |V | |u| ϕ2 ≤ B2r∩{|u|≤1} |V | |u| + B2r∩{|u|1} |V | |u| ϕ2 ≤ B2r |V | + B2r |V | |uϕ| 2 , it follows that V u ∈ L1 (Br). Definition 4.3. We say that u ∈ H2,2 loc (Ω) is a weak solution of (2.6), if for any ψ ∈ C∞ 0 (Ω) we have Ω ∆u(y)∆ψ(y)dy = Ω V (y)u(y)ψ(y)dy. (4.4) The proof of the local boundedness of any weak solution of (2.6) will be proved through a sequence of lemmas. The first one states a Caccioppoli-type estimate([9]). We shall omit its proof Lemma 4.4. Let u be a weak solution of (2.6). If 0 s t and Bt ⊂ Ω, then there exists a constant C = C(V ) independent of u such that Bs |∆u|2 ≤ C (t − s)4 Bt |u|2 . (4.5) Remark 4.5. We note that also the following estimate holds: Bs |∇u|2 ≤ C (t − s)2 Bt u2 . (4.6) where C = C(φV ).
- 27. Harnack Inequality and Biharmonic Equations 15 The following lemma is well known see for instance [7]. Lemma 4.6. Assume that there exist θ ∈ (0, 1), α ≥ 0, a 0 and a real continuous, non-decreasing and positive function I(·) defined on (0, 1] such that I(s) ≤ a 1 (t − s) n I(t) θ , (4.7) holds for all 0 s t ≤ 1. Then, for every s1 0 there exists a constant C = C(θ, n, s1, a) independent of I such that I (s1) ≤ C. (4.8) The next results provide a kind of reversed Hölder inequality. Lemma 4.7. Let B2r be a ball contained in Ω and let u be a weak solution of (2.6). Then, there exists a constant C(φV ) such that 1 |Br/2| Br/2 u2 1/2 ≤ C 1 |Br| Br |u| . (4.9) Proof. Without restriction we may assume that the center of Br is the origin. Let ur(x) = u(rx). Then ur is a solution of ∆2 ur = Vrur in B2(0), where Vr(x) = r4 V (rx) satisfies sup x∈B2 B2 |Vr(y)| |x − y|N−4 dy ≤ sup 0α4r sup w∈Ω Bα(w) |V (z)| χΩ(z) |w − z|N−4 dz. (4.10) Since V ∈ KN,2 loc (Ω), then the right-hand side of (4.10) is bounded and tends to zero as r → 0. We may establish Lemma 4.7 by taking r = 1, Ω = B2, where B2 is centered at the origin. We may also assume that B1 |u| = 1. Define for 0 ≤ s ≤ 1: I(s) = Bs u2 1/2 . Since L1 (Ω) ∩ L 2N N−4 (Ω) → L2 (Ω), we get I(s) ≤ Bs |u| 2N N−4 N−4 2(N+4) . By the Sobolev inequality we obtain for 0 ≤ s t ≤ 1 I(s) ≤ a(φV ) 1 (t − s)2 I(t) N N+4 . (4.11) The conclusion follows from Lemma 4.6.
- 28. 16 G. Caristi and E. Mitidieri Lemma 4.8. Let p ∈ (0, 2), n ≥ 0 and a 0. Let v ∈ L∞ (B1) be nonnegative and sup Bs v2 ≤ a (t − s)n Bt v2 (x)dx, for every 0 s t ≤ 1. Then, there exists a constant C(n, p, a) 0 such that sup B1/2 vp ≤ C B1 vp (x)dx. Proof. We may suppose that v ≡ 0. Putting γ = B1 vp (x)dx −2/p and I(s) = γ Bs v2 , for 0 s 1, we note that sup B1/2 vp ≤ [6n aI (2/3)] B1 vp (x)dx. In order to complete the proof it is sufficient to bound I (2/3) by a constant depending only on n, p and a. To this end let α = 2 2−p and 0 s t ≤ 1. We obtain I(s)α = γα Bs v2−p vp α ≤ γα sup Bs v2 Bs vp α ≤ aγα (t − s) n Bs vp α Bt v2 ≤ a (t − s) n I(t). Finally, putting θ = 1/α and s1 = 2/3, the conclusion follows from (4.7) and (4.8). The following theorem states the local boundedness of weak solutions of (2.6). Theorem 4.9. Let V ∈ KN,2 loc (Ω). Then, for each p ∈ (0, ∞) there exist C = C(p) 0 and r1 = r1(φV ) 0 such that if u is a weak solution of (2.6), then, for each 0 R ≤ R1 and B2R(x0) ⊂ Ω, we have sup BR/2(x0) |u(x)| ≤ C 1 |Br| BR(x0) |u(y)| p dy 1/p . (4.12) Proof. Without restriction we can assume that x0 = 0. Let G(x, y) be the Green function of the Dirichlet problem for equation (2.13) on B2R, see Proposition 2.10. Choose R/2 ≤ s t ≤ R and a function ϕ ∈ C∞ 0 (Bt−(t−s)/4) such that 0 ≤ ϕ ≤ 1, ϕ ≡ 1 on B(t+s)/2, and Dk ϕ ≤ c/(t − s)k , for k = 1, 2. We claim that for almost all x ∈ B2R the following identity holds: u(x)ϕ(x) = B2R ∆yG(x, y) u(y) ∆ϕ(y) dy − B2R G(x, y) ∆u(y) ∆ϕ(y) dy + 2 B2R ∆yG(x, y) (∇u(y), ∇ϕ(y)) dy − 2 B2R (∇yG(x, y), ∇ϕ(y))∆u(y) dy.
- 29. Harnack Inequality and Biharmonic Equations 17 We shall verify this identity assuming that u ∈ C4 (B2R). The general case can be deduced by a standard approximation argument. We have ∆2 (uϕ) = u ∆2 ϕ + 2 (∇∆ϕ, ∇u) + 2∆ϕ ∆u + 2 (∇∆u, ∇ϕ) + 2∆ (∇ϕ, ∇u) + ϕ ∆2 u. Hence, ∆2 (uϕ) − V uϕ = u∆2 ϕ + 2 (∇∆ϕ, ∇u) + 2∆ϕ ∆u + 2 (∇∆u, ∇ϕ) + 2∆ (∇ϕ, ∇u) , and then, u ϕ = G u∆2 ϕ + 2 (∇∆ϕ, ∇u) + 2∆ϕ ∆u + 2 (∇∆u, ∇ϕ) + 2∆ (∇ϕ, ∇u) = ∆G ∆ϕ u − G ∆u ∆ϕ − 2 (∇G, ∇ϕ) ∆u + 2 ∆G (∇u , ∇ϕ) . Let B denote the set Bt−(t−s)/4 B(t+s)/2. Then, for almost all x ∈ B2R we have |u(x)ϕ(x)| ≤ c (t − s)2 B |∆yG(x, y)|2 dy 1/2 Bt u2 (y) dy 1/2 + c (t − s)2 B |G(x, y)| 2 dy 1/2 B |∆u(y)| 2 dy 1/2 + c (t − s) B |∆yG(x, y)|2 dy 1/2 B |∇u(y)|2 dy 1/2 + c (t − s) B |∇yG(x, y)| 2 dy 1/2 B |∆u(y)| 2 dy 1/2 . From Lemma 4.4 it follows that: B |∆u(y)| 2 dy 1/2 ≤ c (t − s)2 Bt u2 (y) dy 1/2 , and B |∇u(y)| 2 dy 1/2 ≤ c (t − s) Bt u2 (y) dy 1/2 . Using again Lemma 4.4 and Lemma 4.7, we find for x ∈ Bs B |∆yG(x, y)| 2 dy 1/2 ≤ c (t − s)5/2 Bt G(x, y) dy, B |∇yG(x, y)|2 dy 1/2 ≤ c (t − s)3/2 Bt G(x, y) dy, and B |G(x, y)| 2 dy 1/2 ≤ c (t − s)1/2 Bt G(x, y) dy.
- 30. 18 G. Caristi and E. Mitidieri Since Bt G(x, y) dy is bounded, from the above estimates we obtain sup Bs |u(x)| ≤ c (t − s)5 Bt u(y)2 dy 1/2 . (4.13) Finally in virtue of Lemma 4.8 and estimate (4.13) we get the conclusion, for p ∈ (0, 2): sup Br/2 |u(x)| ≤ C(p) Br |u(y)| p dy 1/p . The case p ∈ (2, ∞) follows applying Hölder inequality. The continuity of weak solutions can be proved as in the previous section. 4.1. The Hölder continuity The proof of Theorem 4.12 below will be based on two lemmas, which are adap- tations of the Lemmas 3.1 and 3.2 of [23]. The first one is the following. Lemma 4.10. Let 0 β 1 and x0, x ∈ RN , r 0 be such that λ := |x − x0| r 1−β ≤ 1. (4.14) Then, there exists a constant c 0 such that for y with |y − x0| ≥ 2 r1−β |x − x0| β we have 1 |y − x| N−4 − 1 |y − x0| N−4 ≤ c |x − x0| r 1−β 1 |y − x0| N−4 . (4.15) Proof. Let y ∈ RN be such that |y − x0| ≥ 2 r1−β |x − x0|β . (4.16) Since |x − x0| = |x − x0| β |x − x0| 1−β ≤ λ/2 |y − x0|, it follows that |y − x| ≥ |y − x0| − |x − x0| ≥ 1 − λ 2 |y − x0| . From this inequality we get 1 |y − x| N−4 − 1 |y − x0| N−4 = ||y − x0| − |y − x|| N−5 k=0 |y − x0| k−N+4 |y − x| −k−1 ≤ |x − x0| |y − x0| N−3 N−5 k=0 1 1 − λ/2 k+1 . Hence, the conclusion follows with c = N−5 k=0 2k .
- 31. Harnack Inequality and Biharmonic Equations 19 Now, let x0 ∈ Ω, R 0 be such that B4R(x0) ⊂ Ω and let g ∈ KN,2 loc (Ω) be nonnegative with φg(t) = φg(t, B2R(x0)). Let ψ ∈ C∞ 0 (B2R(x0)), 0 ≤ ψ ≤ 1 and J(x) := cN g(y) ψ(y) |y − x| N−4 dy, x ∈ RN . We know that J(·) is continuous on B2R(x0). The next lemma provides an estimate of the local modulus of continuity of J. Lemma 4.11. There exists a constant c 0 such that for any β ∈ (0, 1) and x such that |x − x0| ≤ r we have |J(x) − J(x0)| ≤ c |x − x0| r 1−β φg(2r) + 2φg 3 r1−β |x − x0| β . (4.17) Proof. Without loss of generality we may assume that x0 = 0. Choose 0 β 1 and let 0 |x| r. It follows that |J(x) − J(0)| ≤ cN g(y)χB2r (y) |x − y| 4−N − |y| 4−N dy ≤ cN Br∩{y:|y|≥2r1−β|x|β } · · · + cN Br∩{y:|y|≤2r1−β|x|β } · · · := J1(x) + J2(x). Now, applying Lemma 4.10 to J1 we obtain J1(x) ≤ c |x| r 1−β g(y)χB2r (y) |y| N−4 dy ≤ c |x| r 1−β φg(2r). (4.18) Consider J2(·). Since |y| ≤ 2r1−β |x| β and |x| 1−β ≤ r1−β we deduce that |y − x| ≤ |x| + |y| ≤ |x| β |x| 1−β + 2r1−β ≤ 3r1−β |x| β , and then, J2(x) ≤ cN B2r∩{y:|y−x|≤3r1−β|x|β } g(y) |y − x| N−4 dy + cN B2r∩{y:|y|≤2r1−β|x|β } g(y) |y|N−4 dy ≤ 2ηg 3r1−β |x|β . The conclusion now follows from this inequality and (4.18). Theorem 4.12. Let x0 ∈ Ω and B2R(x0) ⊂ Ω. If u is a weak solution of (2.6), then, chosen 0 β 1 there exists a constant c such that for |x − x0| r we have |u(x) − u(x0)| ≤ ≤ c |x − x0| r 1−β (φV (2r) + 1) + φV 3r |x − x0| r β sup B3r(x0) |u| .
- 32. 20 G. Caristi and E. Mitidieri Proof. Without loss of generality we may assume that x0 = 0 and set B4r = B4r(0). Let ϕ ∈ C∞ 0 (B2r), 0 ≤ ϕ ≤ 1 with ϕ ≡ 1 on B3r/2 and Dk ϕ ≤ c/r|k| , for k = 1, 2. We have u(x)ϕ(x) = Φ2(x, y) V (y) u(y) ϕ(y) dy − Φ2(x, y)∆u(y)∆ϕ(y)dy + ∆yΦ2(x, y) u(y) ∆ϕ(y) dy − 2 ∇yΦ2(x, y) · ∇ϕ(y)∆u(y) dy + 2 ∆yΦ2(x, y)∇u(y) · ∇ϕ(y) dy = 5 i=1 Ii(x). Fix β ∈ (0, 1) and let |x| r. From this identity we obtain u(x) − u(0) = (Φ2(x, y) − Φ2(0, y) ) V (y) u(y) ϕ(y) − (Φ2(x, y) − Φ2(0, y)) ∆u(y)∆ϕ(y)dy + (∆yΦ2(x, y) − ∆yΦ2(0, y)) u(y)∆ϕ(y) − 2 (∇yΦ2(x, y) − ∇yΦ2(0, y)) ∇ϕ(y) ∆u(y) + 2 (∆yΦ2(x, y) − ∆yΦ2(0, y)) ∇u(y)∇ϕ(y). (4.19) Since V u ∈ KN,2 loc (B2r) and φV u(t) ≤ supB2r |u| · φV (t), we can apply Lemma 4.11 to the first term on the right-hand side of (4.19) and obtain |I1(x) − I1(0)| ≤ c |x| r 1−β ηV (2r) + 2ηV 3r1−β |x| β sup B2r |u| . (4.20) To handle the other terms of (4.19), let us observe that all the derivatives of ϕ vanish for |y| ≤ r. Define for t ∈ [0, 1] φ(t) := |y − tx|4−N , where y ∈ B2r B3r/2 and |x| r. From the mean value theorem and |y − t0x| ≥ |y|−|x| ≥ r/2 it follows that φ(1) − φ(0) = |y − x| 4−N − |y| 4−N = φ (t0) ≤ (N − 4) |x| |y − t0x| N−3 ≤ (N − 4) 2N−3 |x| rN−3 , where t0 ∈ (0, 1). Therefore, |Φ2(x, y) − Φ2(0, y)| ≤ c |x| rN−3 ≤ c |x| r 1−β 1 rN−4 . (4.21)
- 33. Harnack Inequality and Biharmonic Equations 21 The same argument shows also that |∇yΦ2(x, y) − ∇yΦ2(0, y)| ≤ c |x| r 1−β 1 rN−3 , (4.22) |∆yΦ2(x, y) − ∆yΦ2(0, y)| ≤ c |x| r 1−β 1 rN−2 . (4.23) Now we can estimate the remaining terms of (4.19), that is |Ik(x) − Ik(0)| for k = 2, . . . , 5, by using the bounds (4.21), (4.22) and (4.23). We find that |I2(x) − I2(0)| ≤ c r2 |Φ2(x, y) − Φ2(0, y)| |∆u(y)| dy ≤ c |x| r 1−β 1 rN−2 B2r |∆u| ≤ c |x| r 1−β r2 1 |B2r| B2r |∆u| 2 1/2 , and hence, by Lemma 4.4 |I2(x) − I2(0)| ≤ c |x| r 1−β 1 |B3r| B3r |u| 2 1/2 . (4.24) It is also easy to check that |I3(x) − I3(0)| ≤ c |x| r 1−β 1 |B3r| B3r |u| 2 1/2 , (4.25) and |I4(x) − I4(0)| + |I5(x) − I5(0)| ≤ c |x| r 1−β 1 |B3r| B3r |u|2 1/2 . Finally, this inequality together with the estimates (4.20), (4.24) and (4.25) com- plete the proof. In order to conclude that the weak solutions of (2.6) are Hölder continuous we restrict ourselves to a class of potentials V such that φV (t) ≤ Mtα (4.26) for some α 0. Clearly, this class is not empty since, according to Example 2.3, we know that if V satisfies a Stummel condition, then (4.26) holds with α = γ/2, or if V belong to Lp loc with p N 4 , then (4.26) holds with α = 4 − N p . We point out that if N ≤ 3 all weak solutions u ∈ H2,2 loc (Ω) of (2.6) are Hölder continuous by Sobolev’s embedding. Theorem 4.13. Let x0 ∈ Ω and B4R(x0) ⊂ Ω. Let u be a weak solution of (1.1). Suppose that there exist α 0 and a constant MV = M(V, B4R) such that (4.26) holds. Then, if |x − x0| ≤ R, we have |u(x) − u(x0)| ≤ c [MV (3R) α + 1] R− α 1+α sup B3R(x0) |u| |x − x0| α 1+α . (4.27)
- 34. 22 G. Caristi and E. Mitidieri Proof. Since φV (2r) ≤ MV (2R) α and φV 3r |x − x0| R β ≤ MV (3r) α |x − x0| R αβ , the conclusion follows directly from Theorem 4.12 by taking β = 1 1+α . 5. Applications In this section we shall present some consequences of Theorem 4.9. The first is another proof of Harnack inequality: Theorem 5.1. Assume that V ∈ KN,2 (Ω). Let u be a nonnegative weak solution of (1.1) such that −∆u ≥ 0 in Ω. Then there exist C = C(N) and r0 = r0(ηV ) such that for each 0 R ≤ R0 and B2R ⊂ Ω we have sup BR/2 u ≤ C inf BR/2 u. Proof. Since u is nonnegative and satisfies −∆u ≥ 0 in Ω, then we have (see, e.g., [11]) inf Br/2 u(x) ≥ C(N) 1 |Br| Br u(y) dy, for all r 0 such that B2r ⊂ Ω. The conclusion follows from this inequality and Theorem 4.9 with p = 1. Theorem 5.2. Let p : 1 ≤ p ∞ and Ω = RN BR(0). If u ∈ Lp (Ω) ∩ H2,2 loc (Ω) is a solution of (1.1) with V ∈ KN,2 (Ω) nonnegative then: lim |x|→∞ |u (x)| = 0 Proof. Without loss of generality we may assume that u is continuous in Ω. By Theorem 4.9, it follows that there exists r0 = r0(ηV ) such that for all x : |x| ≥ R+2, for each r ∈ (0, r0) and 0 s t 1 we have sup Bsr(x) |u(y)|p ≤ C rN (t − s) N Btr(x) |u(z)|p dz. Since u ∈ Lp (Ω) the result follows. The next result concerns the limiting case for a Harnack’s theorem near the possible isolated singularity x = 0 (compare with Theorem 3.1 of [10]). Theorem 5.3. Let Ω = BR(0) {0} and let V ∈ Cloc (Ω) be nonnegative and such that for all x ∈ Ω 0 ≤ V (x) ≤ c1 |x| 4 , c1 0. Let u ∈ H2,2 loc (Ω) be a solution of ∆2 u = V u on Ω, u ≥ 0, −∆u ≥ 0 in Ω. (5.1)
- 35. Harnack Inequality and Biharmonic Equations 23 Then there exist two constants C and ϑ1 depending on c1 such that for each 0 ϑ ≤ ϑ1 and 0 1 2R we have sup ≤|x|≤(1+ϑ) u(x) ≤ C inf ≤|x|≤(1+ϑ) u(x). (5.2) Proof. The proof is a slight modification of the proof of Theorem 4.9. Here, we shall emphasize the dependence of the constants on c1. If x0 ∈ Ω let r0 = 1 4 |x0| so that B2r0 (x0) ⊂ Ω. We claim that for every p ∈ (0, ∞) there exists positive constants C = C(p) and ϑ0 depending on c1 such that for all x0 ∈ Ω and 0 r ≤ ϑ0r0 sup Br/2(x0) |u(x)| ≤ C 1 |Br(x0)| Br(x0) |u(y)| p dy 1/p . (5.3) To prove (5.3) we fix σ ∈ (0, 4) and note that sup x∈Ω rσ V L N 4−σ (Br(x)) = C1 ∞ where C1 depends on c1. This fact together with the embedding property ( 4.2) yields the Lemma 3.2 for each Bsr0 (x0) ⊂ Btr0 (x0) ⊂ Ω. Clearly, Lemma 4.7 and 4.8 hold true. We have to bound sup x∈B2r(x0) B2r(x0) V (y) |x − y| N−4 dy, uniformly with respect to x0 ∈ Ω. Indeed, since sup x∈B2r(x0) B2r(x0) V (y) |x − y| N−4 dy ≤ c rσ 0 V L N 4−σ (B2r0 (x0)) rσ rσ 0 , then sup x∈B2r(x0) B2r(x0) V (y) |x − y| N−4 dy ≤ δ, holds, provided that 0 r ≤ ϑ0r0, where ϑ0 = δ/(2cC1). Hence, it suffices to apply those lemmas in order to obtain (5.3). Combining (5.3) with inf Br/2(x0) u(x) ≥ c |Br(x0)| Br(x0) u(y) dy, corresponding to u ≥ 0 and −∆u ≥ 0, we obtain for 0 r ≤ ϑ0r0 sup Br/2(x0) u(x) ≤ C inf Br/2(x0) u(x). (5.4) Finally, the inequality (5.2) follows from (5.4) by applying a standard covering argument in the annulus ≤ |x| ≤ (1 + ϑ). 5.1. Universal estimates The following lemma is due to Mitidieri and Pohozaev [17]. Lemma 5.4. Let q 1. Assume that u is a weak solution of the problem ∆2 u(x) ≥ |u(x)|q , x ∈ Ω ⊂ RN , (5.5)
- 36. 24 G. Caristi and E. Mitidieri then, there exist C 0 independent of u and R 0 such that BR |u(x)|q dx ≤ CR N(q−1)−4q q−1 , (5.6) where BR = BR(x0) ⊂ Ω. Proof. Choose ψ ∈ C∞ 0 (RN ) radially symmetric such that supp(ψ) = B2(0), 0 ≤ ψ ≤ 1 and ψ ≡ 1 in B1(0). Given x0 ∈ Ω and R 0 such that B2R(x0) ⊂⊂ Ω, define ψR = ψ(R−1 (x−x0)). Multiply (5.5) by ψ and integrate by parts. By Hölder inequality, we get Ω |u|q ψRdx≤ Ω ∆2 uψRdx≤ Ω u∆2 ψRdx≤ Ω |u|q ψRdx 1 q Ω |∆2 ψ|q ψq−1 dx 1 q . By a standard change of variables we obtain: BR |u|q dx ≤ C B1 |∆2 ψ|q ψq−1 dx 1 q R N(q−1)−4q q−1 . This concludes the proof. Theorem 5.5. Let N 4 and Ω = RN . Let f : R → R be a given function such that there exists q ∈ (1, N/(N − 4)) and K 1 such that |u|q ≤ f(u) ≤ K|u|q , u ∈ R (5.7) Then, there exist C1 0 and C2 0 (depending only on N, q and K) such that if u ∈ Lq loc(Ω) is a weak solution of ∆2 u(x) = f(u(x)), x ∈ Ω, (5.8) then, the following estimates hold |u(x)| ≤ C1 d(x, ∂Ω)− 4 q−1 , x ∈ Ω, (5.9) |∆u(x)| ≤ C2 d(x, ∂Ω)− 2(q+1) q−1 , x ∈ Ω. (5.10) Proof. Let x0 ∈ Ω and R0 0 be such that B2R0 (x0) ⊂⊂ Ω. Since q N N−4 , we can fix σ 0 such that q = N N − 4 + σ N N − 4 . It follows that V ≡ |u|q−1 ∈ L N/(4−σ) loc (Ω). In view of Example 2.4 we know that, V ∈ KN,2 loc (Ω). Let x0 ∈ Ω and R0 be chosen as in Theorem 3.3. Take δ0 = R/8. From (5.7) and Lemma 2.7, we get: |u(x)| ≤ C R4−N 0 V u L1(B 3R 2 (x0) + C R−N 0 u L1(B R0 4 +R (x0))
- 37. Harnack Inequality and Biharmonic Equations 25 where C and C1 are positive constants which depend on N. From Lemma 5.4 and Hölder inequality, we get that |u(x)| ≤ C1R4−N 0 R N(q−1)−4q q−1 0 + C2R−N 0 R N(q−1)−4q q(q−1) 0 R N q 0 , (5.11) which implies that for any x ∈ BR(x0) |u(x)| ≤ C R− 4 q−1 . Thus (5.9) follows by choosing R = d(x, ∂Ω). A similar argument as above can be used to show that indeed (5.10) holds. We omit the details. Acknowledgment We thank the referee for useful comments. References [1] M. Aizenman and B. Simon, Brownian Motion and Harnack Inequality for Schrö- dinger Operators, Commun. Pure Appl. Math., 35 (1982), 209–273. [2] N. Aronszajn, T.M. Creese, L.J. Lipkin, Polyharmonic functions, Clarendon Press, Oxford, 1983. [3] I. Bachar, H. Mâagli, and L. Mâatoug, Positive solutions of nonlinear elliptic equations in a half-space in R2 , Electronic Journal of Differential Equations, Vol. 2002(2002), No. 41, pp. 1–24. [4] I. Bachar, H. Mâagli, and M. Zribi, Existence of positive solutions to nonlinear el- liptic problem in the half space, Electronic Journal of Differential Equations, Vol. 2005(2005), No. 44, pp. 1–16. [5] G. Caristi, E. Mitidieri and R. Soranzo, Isolated Singularities of Polyharmonic Equa- tions, Atti del Seminario Matematico e Fisico dell’Università di Modena, Suppl. al Vol. XLVI (1998), pp. 257–294. [6] Z.-Q. Chen, Z. Zhao, Harnack principle for weakly coupled elliptic systems, J. Diff. Eq. 139 (1997), 261–282. [7] F. Chiarenza, E. Fabes, N. Garofalo, Harnack’s inequality for Schrödinger operators and the continuity of solutions, Proc. Amer. Math. Soc. 98 (1986), 415–425. [8] E.B. Fabes and D.W. Stroock, The Lp -integrability of Green’s function and fun- damental solutions for elliptic and parabolic equations, Duke Math. J., 51 (1984), 997–1016. [9] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, Princeton University Press, 1983. [10] B. Gidas, J. Spruck, Global and Local behavior of Positive Solutions of Nonlinear Elliptic Equations, Comm. Pure Appl. Math., 34 (1981), 525–598. [11] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, Springer Verlag, 1983. [12] H.-Ch. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math.Ann. 307 (1997), 589–626.
- 38. 26 G. Caristi and E. Mitidieri [13] A.M. Hinz, H. Kalf, Subsolution estimates and Harnack inequality for Schrödinger operators, J. Reine Angew. Math., 404 (1990), 118–134. [14] H. Mâagli, F. Toumi, M. Zribi, Existence of positive solutions for some polyhar- monic nonlinear boundary-value problems, Electronic Journal of Differential Equa- tions, Vol. 2003(2003), No. 58, pp. 1–19. [15] F. Mandras, Diseguaglianza di Harnack per i sistemi ellittici debolmente accoppiati, Bollettino U.M.I., 5 14-A (1977), 313–321. [16] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in RN , Differential and Integral Equations, Vol. 9, N. 3, (1996), 465–479. [17] E. Mitidieri, S.I. Pohozaev, A priori estimates and nonexistence of solutions to non- linear partial differential equations and inequalities, Tr. Math.Inst. Steklova, Ross. Akad. Nauk, vol. 234, 2001. [18] M. Nicolescu, Les Fonctions Polyharmoniques, Hermann and Cie Editeurs, Paris 1936. [19] M. Schechter, Spectra of partial differential operators, North-Holland, 1986. [20] J. Serrin, Isolated Singularities of Solutions of Quasi-Linear Equations, Acta Math., 113 (1965), 219–240. [21] J. Serrin, Local behavior of Solutions of Quasi-Linear Equations, Acta Math., 111 (1964), 247–302. [22] J. Serrin, H. Zou, Cauchy-Liouville and universal boundedness theorems for quasi- linear elliptic equations and inequalities, Acta Math., 189 (2002), 79–142. [23] Ch.G. Simader, An elementary proof of Harnack’s inequality for Schrödinger opera- tors and related topics, Math. Z. 203 (1990), 129–152. [24] Ch.G. Simader, Mean value formula, Weyl’s lemma and Liouville theorems for ∆2 and Stoke’s System, Results Math. 22 (1992), 761–780. [25] R. Soranzo, Isolated Singularities of Positive Solutions of a Superlinear Biharmonic Equation, Potential Analysis, 6 (1997), 57–85. [26] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189–258. [27] N.S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic partial differential equations, Comm. Pure Appl. Math., 20 (1967), 721–747. [28] Z. Zhao, Conditional gauge with unbounded potential, Z. Wahrsch. Verw. Gebiete, 65 (1983), 13–18. [29] Z. Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl., 116 (1986), 309–334. [30] Z. Zhao, Uniform boundedness of conditional gauge and Schrödinger equations, Com- mun. Math. Phys., 93 (1984), 19–31. Gabriella Caristi and Enzo Mitidieri Dipartimento di Matematica e Informatica Università di Trieste I-34127 Trieste, Italy e-mail: caristi@univ.trieste.it e-mail: mitidier@univ.trieste.it
- 39. Operator Theory: Advances and Applications, Vol. 168, 27–43 c 2006 Birkhäuser Verlag Basel/Switzerland Clément Interpolation and Its Role in Adaptive Finite Element Error Control Carsten Carstensen In honor of the retirement of Philippe Clément. Abstract. Several approximation operators followed Philippe Clément’s sem- inal paper in 1975 and are hence known as Clément-type interpolation op- erators, weak-, or quasi-interpolation operators. Those operators map some Sobolev space V ⊂ W k,p (Ω) onto some finite element space Vh ⊂ W k,p (Ω) and generalize nodal interpolation operators whenever W k,p (Ω) ⊂ C0 (Ω), i.e., when p ≤ n/k for a bounded Lipschitz domain Ω ⊂ Rn . The original motivation was H2 ⊂ C0 (Ω) for higher dimensions n ≥ 4 and hence nodal interpolation is not well defined. Todays main use of the approximation operators is for a reliability proof in a posteriori error control. The survey reports on the class of Clément type interpolation operators, its use in a posteriori finite element error control and for coarsening in adaptive mesh design. 1. Introduction: Motivation and applications The finite element method (FEM) is the driving force and dominating tool behind today’s computational sciences and engineering. It is common sense that the FEM should be implemented in an adaptive mesh-refining version with a posteriori error control. This paper high-lights one mathematical key tool in the justification of those adaptive finite element methods (AFEMs) which dates back to Philippe Clément’s seminal paper in 1975. The origin of his operators, today known under the description Clément-type interpolation operators, weak-, or quasi-interpolation operators in the literature, however, was completely different. The author is supported by the DFG Research Center Matheon “Mathematics for key technolo- gies” in Berlin.
- 40. 28 C. Carstensen Indeed, many second-order elliptic boundary value problems are recast in a weak form a(u, v) = b(v) for all v ∈ V, where (V, a) is some Hilbert space with induced norm · a and b ∈ V ∗ has the Riesz representation u ∈ V . Given a subspace Vh, the discrete solution uh ∈ Vh satisfies a(uh, vh) = b(vh) for all vh ∈ Vh. The error e := u − uh ∈ V then satisfies the best-approximation property e a = min vh∈Vh u − vh a. (The proof is given at the end of Subsection 3.1 below for completeness.) The classical estimation of the upper bound u − vh a replaces vh by the nodal interpolation operator applied to the exact solution u. In the applications, one typically has u ∈ V ∩ Hs (Ω; Rm ) for some Sobolev space Hs (Ω; Rm ) on some domain Ω and some regularity pa- rameter s ≥ 1. The higher regularity with s 1 is subject to regularity theory of elliptic PDEs and s depends on the smoothness of data and coefficients as well as on the boundary conditions and on the geometry (e.g., corners) of Ω. An optimal regularity for 2D and convex domains or C2 boundaries is s = 2. For smaller s and/or for higher space dimensions, there holds u ∈ V ∩ Hs (Ω; Rm ) ⊂ C(Ω; Rm ). Thus the nodal interpolation operator, which evaluates the function u at a finite number of points, is not defined well. The Clément operators J(v) cures that difficulty in that they replace the evaluation at discrete points by some initial local best approximation of v in the L2 sense followed by the point evaluation of this local best approximation at the discrete degrees of freedom. This small modification is essential in order to receive approximation and stability properties in the sense that v − J(v) Hr (Ω) ≤ C hs−r v Hs(Ω) with some mesh-size independent constant C 0 and the maximal mesh-size h 0. The point is that the range for the exponents s and r is possible with r = 0, 1 and s = 1, 2 while s = 1 is excluded for the nodal interpolation operator even for dimension n ≥ 2. At the time of Clément’s research, this improvement seemed to be regarded as a marginal technicality in the a priori error control – this possible underestima- tion is displayed in Ciarlet’s finite element book [17] where Clément’s research is summarized as an exercise (3.2.3).
- 41. Clément Interpolation in AFEM 29 The relevance of Clément’s results was highlighted in the a posteriori er- ror analysis of the last two decades where the aforementioned first-order ap- proximation and stability property of J for r ≤ s = 1 had been exploited. Amongst the most influential pioneering publications on a posteriori error con- trol are [2, 21, 5, 19] followed by many others. The readers may find it rewarding to study the survey articles of [20, 7] and the books of [26, 1, 3, 4] for a first insight and further references. This paper gives a very general approach to Clément-type interpolation in Section 2 and comments on its applications in the a posteriori error analysis in Section 3 where further applications such as the error reduction in adaptive finite element schemes and the coarsening are seen as currently open fields. 2. Clément-type approximation This section aims at a first glance of a class of approximation operators J : W1,p (Ω) → W1,p (Ω) with discrete image which are named after Philippe Clément and called Clément-type interpolation operators or sometimes quasi-interpolation operators. The most relevant properties include the local first order approximation, i.e., for all v ∈ V ⊂ W1,p (Ω) with weak gradient Dv ∈ Lp (Ω)n in Ω ⊂ Rn and an underlying triangulation with local mesh-size h ∈ L∞ (Ω) and discrete space Vh, there holds v − J(v) Lp(Ω) ≤ C hDv Lp(Ω) and h−1 (v − J(v)) Lp(Ω) ≤ C Dv Lp(Ω), and the W1,p (Ω) stability property, i.e., for v ∈ V , DJ(v) Lp(Ω) ≤ C Dv Lp(Ω). A discussion on a particular operator with an additional orthogonality property ends this section. Although the emphasis in this section is not on the evaluation of constants and their sharp estimation, there is a list M1, . . . , M5 of highlighted relevant parameters. Definition 2.1 (abstract assumptions). Let Ω be a bounded Lipschitz domain in Rn , let 1 p, q ∞ with 1/p + 1/q = 1 and m, n ∈ N = {1, 2, 3, . . .}. Suppose that (ϕz : z ∈ N) is a Lipschitz continuous partition of unity [associated to Vh below] on Ω with ωz := {x ∈ Ω : ϕ(x) = 0} ⊆ Ωz ⊂ Ω. The sets Ωz are supposed to be open, nonvoid, and connected supersets of ωz of volume |Ωz|. For any z ∈ N let Az ⊆ Rm be nonvoid, convex, and closed and let Πz : Rm → Rm be the orthogonal projection onto Az in Rm (with respect to the Euclidean metric). Let V ⊆ W1,p (Ω; Rm ) and Vh := { z∈N azϕz : az ∈ Az}.
- 42. 30 C. Carstensen For any z ∈ N denote Vz := V |Ωz := {v|Ωz : v ∈ V } ⊆ W1,p (Ωz; Rm ) and let Hz 0 and ez 0. Suppose there exists a map Jz : Vz → Rm such that the following two hypothesis (H1)–(H2) hold for all z ∈ N and for all v ∈ Vz with v̄z := |Ωz|−1 Ωz v(x) dx ∈ Rm : (H1) v − Πz(vz) Lp(Ωz) ≤ Hz Dv Lp(Ωz ); (H2) Jz(v|Ωz ) − v̄z Lp(Ωz ) ≤ ezHz Dv Lp(Ωz). Typically, (H1) describes some Poincaré-Friedrichs inequality with Hz diam(Ωz) where Ωz is some local neighborhood of a node. The subsequent list of parameters illustrates constants crucial in the analysis of the approximation and stability estimates. Definition 2.2 (Some constants). Under the assumptions of the previous definition set (the piecewise constant function) H(x) := max z∈N x∈Ωz Hz for x ∈ Ω and, with 1/p + 1/q = 1, define M1, . . . , M5 by M1 := max z∈N ϕz L∞(Ω), M2 := max x∈Ω z∈N |ϕz(x)|, M3 := max x∈Ω card{z ∈ N : x ∈ Ωz}, M4 := max z∈N (1 + ez), M5 := sup x∈Ω z∈N H(x)q |Dϕz(x)|q 1/q . Example 2.3 (P1FEM in 2D). Given a regular triangulation T of the unit square Ω = (0, 1)2 into triangles, m = 1, n = 2, let V := H1 0 (Ω), p = q = 2, and let (ϕz : z ∈ N) denote the nodal basis functions of all the nodes with respect to the first-order finite element space. The boundary conditions are described by the sets (Az : z ∈ N). For each node z in the interior, written z ∈ N ∩ Ω, Az = R, while Az := {0} for z on the boundary, written z ∈ N ∩ ∂Ω, ΓD = ∂Ω. Let Jz denote the integral mean operator Jz(v) := |ωz|−1 ωz v(x) dx for v ∈ Vz ⊆ Lp (Ωz) on the patch ωz = Ωz = {x ∈ Ω : ϕ(x) 0} with ez = 0 in (H2). The Dirichlet boundary conditions are then prescribed by (v ∈ Rm ) Πz(v) := v if z ∈ N ∩ Ω, 0 if z ∈ N ∩ ∂Ω.
- 43. Clément Interpolation in AFEM 31 There holds (H1) with Poincaré and Friedrich’s inequalities and Hz := cP (ωz) if z ∈ N ∩ Ω, cF (ωz, (∂Ω) ∩ (∂ωz)) if z ∈ N ∩ ∂Ω; Hz is of the form global constant times diam(ωz). It is always true that M1 = M2 = 1 = M4 and M3 = 3. Since Dϕz L∞(Ω) ≈ H−1 z there holds M5 1. Notice that M5 may be very large for small angles in the triangulation while the constants M1, . . . , M4 are robust with respect to small aspect ratios. The aforementioned example is not exactly the choice of [18] but certainly amongst the natural choices [14, 25]. The announced approximation and stability properties are provided by the following main theorem. Theorem 2.4. The map J : V → Vh; v → z∈N Πz(Jz(v|Ωz ))ϕz satisfies (a) ∃c1 0 ∀v ∈ V, v − J(v) Lp(Ω) ≤ c1 HDv Lp(Ω); (b) ∃c2 0 ∀v ∈ V, H−1 (v − J(v)) Lp(Ω) ≤ c2 Dv Lp(Ω), (c) ∃c3 0 ∀v ∈ V, DJ(v) Lp(Ω) ≤ c3 Dv Lp(Ω). The constants c1 and c2 depend on M1, . . . , M4 and on p, q, m, n while c3 depends also on M5. Proof. Given v ∈ V , let vz := Jz(v|Ωz ), z ∈ N, be the coefficient in J(v) = z∈N Πz(vz)ϕz and let vz := |Ωz|−1 Ωz v(x) dx be the local integral mean. The first step of this proof establishes the estimate v − Πz(vz) Lp(Ωz ) ≤ M4Hz Dv Lp(Ωz ). (2.1) The assumptions (H1) and (H2) lead to v|Ωz − Πz(vz) Lp(Ωz) ≤ Hz Dv Lp(Ωz ), vz − vz Lp(Ωz) ≤ ezHz Dv Lp(Ωz ). Since Πz is non-expansive (Lipschitz with Lip(Πz) ≤ 1; Rm is endowed with the Euclidean norm used in Lp (Ω; Rm )) there holds Πz(vz) − Πz(vz) Lp(Ωz ) ≤ vz − vz Lp(Ω). The combination of the aforementioned estimates yields an upper bound of the right-hand side in v − Πz(vz) Lp(Ωz) ≤ v − Πz(vz) Lp(Ωz ) + Πz(vz) − Π(vz) Lp(Ωz ) and, in this way, establishes (2.1) and concludes the first step.
- 44. 32 C. Carstensen Step two establishes assertion (a) of the theorem. Since (ϕz)z∈N is a partition of unity, z∈N ϕz = 1 in Ω, there holds v − J(v) p Lp(Ω) = z∈N (v − Πz(vz))ϕz p Lp(Ω) = Ω z∈N ϕ1/q z ϕ1/p z (v − Πz(vz)) p dx. Hölders’s inequality in 1 and z∈N |ϕz| ≤ M2 lead to v − J(v) p Lp(Ω) ≤ Ω z∈N |ϕz| |v − Πz(vz)|p z∈N |ϕz| p/q dx ≤ M p/q 2 z∈N Ω |ϕz| |v − Πz(vz)|p dx ≤ M1M p/q 2 z∈N v − Πz(vz) p Lp(Ωz ). This and estimate (2.1) from the first step show assertion (a) with c1 := M 1/p 1 M 1/q 2 M 1/p 3 M4. In fact, the last step involves the overlaps by means of M3 and the definition of H(x) := max{Hz : ∃z ∈ N, x ∈ Ωz}: z∈N Hp z Dv p Lp(Ωz) ≤ Ω z∈N x∈Ωz Hp z |Dv(x)|p dx ≤ Ω H(x)p ( z∈N x∈Ωz 1)|Dv(x)|p dx ≤ M3 H Dv p Lp(Ω). This concludes step two and the proof of assertion (a). Step three establishes assumption (b) of the theorem. The same list of argu- ments as in step two shows H−1 (v − J(v)) p Lp(Ω) ≤ M1M p/q 2 z∈N H−1 (v − Πz(vz)) p Lp(Ωz ). Since Hz ≤ H(x) for x ∈ Ωz, z ∈ N, there holds H−1 (v − J(v)) p Lp(Ω) ≤ M1M p/q 2 z∈N H−p z v − Πz(vz) p Lp(Ωz). Based on (2.1), the proof of assertion (b) and step three are concluded as in step two. This yields (b) with c2 := M1/p M 1/q 2 M 1/p 3 M4.
- 45. Clément Interpolation in AFEM 33 Step four establishes assumption (c) of the theorem. Since z∈N ϕz = 1 there holds z∈N Dϕz = 0 almost everywhere in Ω. With the characteristic function χz ∈ L∞ (Ω) of Ωz, defined by χz|Ωz = 1 and χz|ΩΩz = 0 it follows that DJ(v) p Lp(Ω) = Ω z∈N χz(x)H−1 (x)(Πz(vz) − v(x))Dϕz(x)H(x) p dx. Hölder’s inequality in 1 shows, for almost every x ∈ Ω, z∈N χz(x)H−1 (x)(Πz(vz) − v(x))Dϕz(x)H(x) ≤ z∈N χz(x)H−p (x)|Πz(vz) − v(x)|p 1/p z∈N H(x)q |Dϕz(x)|q 1/q ≤ M5 z∈N χz(x)H−p z |Πz(vz) − v(x)|p 1/p . The combination of the two preceding estimates in this step four is followed by (2.1) and then results in DJ(v) p Lp(Ω) ≤ Mp 4 Mp 5 z∈N Dv p Lp(Ωz). The proof concludes as in step two and establishes assertion (c) with c3 := M 1/p 3 M4M5. In order to discuss a particular operator designed to introduce an extra or- thogonality property, additional assumptions are necessary. Definition 2.5 (Free nodes for Dirichlet boundary conditions). Adopt the notation of Definition 2.1 and suppose, in addition, that Vh ⊂ V := {v ∈ W1,p (Ω; Rm ) : v = 0 on ΓD}, where ΓD is some (possibly empty) closed part of the boundary that includes a complete set of edges in the sense that, for each E ∈ E, either E ⊂ ΓD or E∩ΓD ⊂ N. Let K := N ΓD and, for all z ∈ N, Az = Rm if z ∈ K; {0} otherwise. Given any z ∈ K let J (z) ⊂ N such that z ∈ J (z) and ψz := x∈J (z) ϕx
- 46. 34 C. Carstensen defines a Lipschitz partition (ψz : z ∈ K) of unity with Ωz ⊇ {x ∈ Ω : ψz(x) = 0}. Then Jz(v) := Ωz vψz dx Ωz ϕz dx defines an approximation operator J : V → Vh. Set M6 := max z∈K ϕz −1 L1(Ωz) ψz Lq(Ωz ) w Lp(Ωz )|Ωz|1/p . Theorem 2.6. The operator J : V → Vh from Definition 2.5 satisfies (a)–(c) from Theorem 2.4 plus the orthogonality property Ω f · (v − Jv) dx ≤ c4 Dv Lp(Ω) z∈K Hq z min fz ∈Rm f − fz q Lq(Ωz)) 1/q . Remark 2.7. A Poincaré inequality in case f ∈ W1,q (Ω), f − fz Lq(Ωz) ≤ Cp(Ωz)Hz Df Lq(Ωz ), illustrates that the right-hand side in Theorem 2.6 is of the form O( H 2 L∞(Ω)) Dv Lp(Ω) and then of higher order (compared with the error of first-order FEM). Proof of Theorem 2.6. In order to employ Theorem 2.4, it remains to check the conditions (H1)–(H2). For any v ∈ Vz = V |Ωz , (H1) is either a Poincaré or Friedrichs inequality for z ∈ K or z ∈ K ∩ ΓD as in Example 2.3. Moreover, given any v ∈ Vz and Jz(v) := Ω vψz dx/ Ω ϕz dx ∈ Rm , there holds Jz(vz) = vz if ψz ≡ ϕz (i.e., J (z) = {z}) for vz := |Ωz|−1 Ωz v(x) dx. Then, for w = v|Ωz − vz, there holds Jz(v|Ωz ) − vz Lp(Ωz) = Jz(w) Lp(Ωz ) ≤ ϕz −1 L1(Ωz) ψz Lq(Ωz ) w Lp(Ωz )|Ωz|1/p ≤ M6 w Lp(Ωz) ≤ M6Hz Dv Lp(Ωz ). For ψz ≡ ϕz (i.e., J (z) = {z}), Ωz includes nodes at the boundary and ∂Ωz ∩ ΓD has positive surface measure. Therefore, Friedrichs inequality guarantees vz Lp(Ωz ) ≤ c5Hz Dvz Lp(Ωz) for all vz ∈ Vz. Then (H2) follows immediately from this and Jz(v) Lp(Ωz ) + vz Lp(Ωz ) ≤ c6 vz Lp(Ωz). The orthogonality property is based on the design of Jz(v|Ωz ) and the partition of unity (ψz : z ∈ K). In fact, Ωz (v(x)ψz(x) − Jz(v|Ωz )ϕz(x)) dx = 0
- 47. Clément Interpolation in AFEM 35 for any z ∈ K and so, for any fz ∈ Rm , there follows Ω f(v − Jv) dx = Ω f(x) z∈K v(x)ψz(x) − z∈K Jz(v|Ωz )ϕz(x) dx = z∈K Ωz f(x) · (v(x)ψz(x) − Jz(v|Ωz )ϕz(x)) dx = z∈K Ωz (f(x) − fz) · (v(x)ψz(x) − Jz(v|Ωz )ϕz(x)) dx ≤ z∈K Hz f − fz Lq(Ωz ) H−1 z vψz − Jz(v|Ωz )ϕz Lp(Ωz ) ≤ z∈K Hq z f − fz q Lq(Ωz ) 1/q z∈K H−p z vψz − Jz(v|Ωz )ϕz p Lp(Ωz ) 1/p . For ψz ≡ ϕz (i.e., J (z) = {z}) it follows v(ψz − ϕz) Lp(Ωz ) + (v − Jz(v|Ωz ))ϕz Lp(Ωz) ≤ c7Hz Dv Lp(Ωz ) as above. For ψz ≡ ϕz, a Friedrichs inequality shows v Lp(Ω) ≤ c8Hz Dv Lp(Ωz ) and completes the proof. The remaining details are similar to the arguments of Theorem 2.4 and hence omitted. 3. A posteriori residual control This section reviews explicit residual-based error estimators in a very abstract form. The subsequent benchmark example motivates this analysis of the dual norm of a linear functional. 3.1. Model example For the benchmark example suppose that an underlying PDE yields to some bi- linear form a on some Sobolev space V such that (V, a) is Hilbert space and the right-hand side b is a bounded linear functional, written b ∈ V ∗ . The Riesz repre- sentation u of b is called weak solution and is characterized by u ∈ V with a(u, v) = b(v) for all v ∈ V. The finite element solution uh belongs to some finite-dimensional subspace Vh of V , the finite element space, and is characterized by uh ∈ Vh with a(uh, vh) = b(vh) for all vh ∈ Vh. In other words, uh is the Riesz representation of the restricted right-hand side b|Vh in the discrete Hilbert space (Vh, a|Vh×Vh ).
- 48. 36 C. Carstensen The goal of a posteriori error control is to represent the error e := u − uh ∈ V in the energy norm · a = a(·, ·). One particular property is the Galerkin orthogonality a(e, vh) = 0 for all vh ∈ Vh. This followed by a Cauchy inequality yields, for all vh ∈ Vh, that e 2 a = a(e, u − vh) ≤ e a u − vh a. This implies the best-approximation property e a = min vh∈Vh u − vh a claimed in the introduction. 3.2. Error and residual Given a, b, and uh, the residual R := b − a(uh, ·) ∈ V ∗ is a bounded linear functional in V , written R ∈ V ∗ . Its dual norm is R V ∗ := sup v∈V {0} R(v)/ v a = sup v∈V {0} a(e, v)/ v a = e a ∞. (3.1) The first equality in (3.1) is a definition, the second equality immediately follows from R = a(e, ·). Notice that the Galerkin orthogonality immediately translates into Vh ⊂ kern R. A Cauchy inequality with respect to the scalar product a results in R V ∗ ≤ e a and hence in one of two inequalities of the last identity R V ∗ = e a in (3.1). The remaining converse inequalities follows in fact with v = e yields finally the equality; the proof of (3.1) is finished. The identity (3.1) means that the error (estimation) in the energy norm is equivalent to the (computation of the) dual norm of the given residual. Further- more, it is even of comparable computational effort to compute an optimal v = e in (3.1) or to compute e. The proof of (3.1) yields even a stability estimate: Given any v ∈ V with v a = 1, the relative error of R(v) as an approximation to e a equals e a − R(v) e a = 1/2 v − e/ e a 2 a. (3.2)
- 49. Clément Interpolation in AFEM 37 In fact, given any v ∈ V with v a = 1, the identity (3.2) follows from 1 − a(e/ e a, v) = 1 2 a(e/ e a, e/ e a) − a(e/ e a, v) + 1 2 a(v, v) = 1 2 v − e/ e a 2 a. The interpretation of the error estimate (3.2) is as follows. The maximizing v in (3.1) (i.e., v ∈ V with maximal R(v) subject to v a ≤ 1) is unique and equals e/ e a. As a consequence, the computation of the maximizing v in (3.1) is equivalent to and indeed equally expensive as the computation of the unknown e/ e a and so essentially of the exact solution u. Therefore, a posteriori error analysis aims to compute lower and upper bounds of R V ∗ instead of its exact value. However, the use of any good guess of the exact solution from some postprocessing with extrapolation or superconvergence phenomenon is welcome (and, e.g., may enter v). 3.3. Residual representation formula Without stating examples in explicit form, it is stressed that linear and nonlinear boundary value problems of second order elliptic PDEs typically lead (after some partial integration) to some explicit representation of the residual R := b−a(uh, ·). Let T denote the underlying regular triangulation of the domain and let E denote a set of edges for n = 2. Then, the data and the discrete solution uh lead to given quantities rT ∈ Rm , the explicit volume residual, and rE ∈ Rm , the jump residual, for any element domain T and any edge E such that the residual representation formula R(v) = T ∈T T rT · v dx − E∈E E rE · v ds (3.3) holds for all v ∈ V . The goal of a posteriori error control is to establish guaranteed lower and upper bounds of the (energy norm of the) error and hence of the (dual norm of the) residual. Any choice of v ∈ V {0} immediately leads to a lower error bound via η := |R(v)/ v a| ≤ e a. Indeed, edge and element-oriented bubble functions are studied for v in [26] to prove efficiency. More challenging appears reliability, i.e., the design of upper error bounds which require extra conditions. In fact, the Galerkin orthogonality Vh ⊂ kern R is supposed to hold for the data in (3.3). 3.4. Explicit residual-based error estimators Given the explicit volume and jump residuals rT and rE in (3.3), one defines the explicit residual-based estimator η2 R := T ∈T h2 T rT 2 L2(T ) + E∈EΩ hE rE 2 L2(E), (3.4)
- 50. 38 C. Carstensen which is reliable in the sense that e a ≤ C ηR. (3.5) The proof of (3.5) follows from the Clément-type interpolation operators and their properties. In fact, given any v ∈ V and any vh ∈ Vh set w := v − vh. Then the residual representation formula with Vh ⊂ kern R plus Cauchy inequalities lead to R(v) = R(w) = T ∈T T rT · w dx − E∈EΩ E rE · w ds ≤ T ∈T hT rT L2(T ) h−1 T w L2(T ) + E∈EΩ h 1/2 E rE L2(E) h −1/2 E w L2(E) ≤ T ∈T h2 T rT 2 L2(T ) 1/2 T ∈T h−2 T w 2 L2(T ) 1/2 + E∈EΩ hE rE 2 L2(E) 1/2 E∈EΩ h−1 E w 2 L2(E) 1/2 . The well-established trace inequality for each element T of diameter hT and its boundary ∂T shows h−1 T w 2 L2(∂T ) ≤ CT h−2 T w 2 L2(T ) + Dw 2 L2(T ) with some size-independent constant CT (which solely depends on the shape of the element domain). This allows an estimate of E∈EΩ h−1 E w 2 L2(E) 1/2 ≤ C T ∈T h−2 T w 2 L2(T ) + Dw 2 L2(T ) . The localized first-order approximation and stability property of the Clément- interpolation shows T ∈T h−2 T w 2 L2(T ) + Dw 2 L2(T ) ≤ C T ∈T Dv 2 L2(T ). Altogether, it follows that R(v) ≤ CηR |v|H1(Ω). This proves reliability in the sense of (3.5) 3.5. Error reduction and discrete residual control Different norms of the residual play an important role in the design of convergent adaptive meshes. Suppose Vh ≡ V is the finite element space with a discrete solution uh ≡ u and its residual R ≡ R := b − a(u , ·). The design task in adaptive finite element methods is to find a superspace V +1 ⊃ V associated with
- 51. Clément Interpolation in AFEM 39 some finer mesh such that the discrete solution u +1 in V +1 ⊂ V is strictly more accurate. To make this precise suppose that 0 1 and 0 ≤ δ satisfy error reduction in the sense that u − u +1 2 a ≤ u − u 2 a + δ. (3.6) The interpretation is that the error is reduced by a factor at most 1/2 1 up to the terms δ which describe small effects. The error reduction leads to discuss the equivalent discrete residual control (1 − ) R 2 V ∗ ≤ R 2 V ∗ +1 + δ. (3.7) Therein, R V ∗ +1 denotes the dual norm of the restricted functional R|V+1 , namely R V ∗ +1 := sup v∈V+1{0} R(v)/ v a. The point is that (3.6)⇔(3.7). The proof follows from (3.1) which leads to R V ∗ = u − u a and R V ∗ +1 = u +1 − u a. This plus the Galerkin orthogonality and the Pythagoras theorem u − u 2 a = u − u +1 2 a + u +1 − u 2 a show that (3.7) reads (1 − ) u − u 2 a ≤ u − u 2 a − u − u +1 2 a + δ and this is (3.6). Altogether, (3.6)⇔(3.7) and one needs to study the discrete residual control. One standard technique is the bulk criterion for the explicit error estimator ηR followed by local discrete efficiency. The latter involves proper mesh-refinement rules and hence this is not included in this paper. 3.6. Remarks This subsection lists a few short comments on various tasks in the a posteriori error analysis. 3.6.1. Dominating edge contributions. For first-order finite element methods in simple situations of the Laplace, Lamé problem, or the primal formulation of elastoplasticity, the volume term rT = f can be substituted by the higher-order term of oscillations, i.e., e 2 a ≤ C osc(f)2 + E∈EΩ hE rE 2 L2(E) . (3.8) Therein, for each node z ∈ N with nodal basis function ϕz and patch ωz := {x ∈ Ω : ϕz(x) = 0} of diameter hz and the source term f ∈ L2 (Ω)m with integral mean fz := |ωz|−1 ωz f(x) dx ∈ Rm the oscillations of f are defined by osc(f) := z∈N h2 z f − fz 2 L2(ωz) 1/2 .
- 52. 40 C. Carstensen Notice for f ∈ H1 (Ω)m and the mesh-size hT ∈ P0(T ) there holds osc(f) ≤ C h2 T Df L2(Ω) and so osc(f) is of quadratic and hence of higher order. We refer to [10, 9, 16], [22], [6], and [24] for further details on and proofs of (3.8). The proof in [16] is based on Theorem 2.6. In fact, the arguments of Subsection 3.4 are combined with rT = f for the lowest-order finite element method at hand. Then, Theorem 2.6 shows Ω fw dx ≤ c4 Dv L2(Ω) osc(f) and leads to the announced higher-order term. Dominating edge contributions lead to efficiency of averaging schemes [10, 11]. 3.6.2. Reliability constants. The global constants in the approximation and sta- bility properties of Clément-type operators enter the reliability constants. The estimates of [14] illustrate some numbers as local eigenvalue problems. Based on numerical calculations, there is proof that, for all meshes with right-isosceles tri- angles and the Laplace equation with right-hand side f ∈ L2 (Ω), there holds reliability in the sense of D(u − uh) L2(Ω) ≤ T ∈T h2 T f 2 L2(T ) 1/2 + E∈E hE E [∂uh/∂νE]2 ds 1/2 with an undisplayed constant Crel 1 in the upper bound [15]. The aspect of upper bounds with constants and corresponding overestimation is empirically discussed in [13]. 3.6.3. Coarsening. One actual challenging detail is coarsening. Reason number one is that coarsening cannot be avoided for time evolving problems to capture moving singularities. Reason number two is the design of algorithms which are guaranteed to convergence in optimal complexity [8]. Any coarsening excludes the Pythagoras theorem (which assumes V ⊂ V +1) and so perturbations need to be controlled which leads to the estimation of min v+1∈V+1 v +1 − u a. Ongoing research investigates the effective estimation of this term. 3.6.4. Other norms and goal functionals. The previous sections concern estima- tions of the error in the energy norm. Other norms are certainly of some interest as well as the error with respect to a certain goal functional. The later is some given bounded and linear functional : V → R with respect to which one aims to monitor the error. That is, one wants to find computable lower and upper bounds for the (unknown) quantity |(u) − (uh)| = |(e)|.
- 53. Clément Interpolation in AFEM 41 Typical examples of goal functionals are described by L2 functions, e.g., (v) = Ω v dx ∀v ∈ V for a given ∈ L2 (Ω) or as contour integrals. To bound or approximate J(e) one considers the dual problem a(v, z) = (v) ∀v ∈ V (3.9) with exact solution z ∈ V (guaranteed by the Lax-Milgram lemma) and the dis- crete solution zh ∈ Vh of a(vh, zh) = (vh) ∀vh ∈ Vh. Set f := z − zh. Based on the Galerkin orthogonality a(e, zh) = 0 one infers (e) = a(e, z) = a(e, z − zh) = a(e, f). (3.10) Cauchy inequalities lead to the a posteriori estimate |(e)| ≤ e a f a ≤ ηuηz. Indeed, utilizing the primal and dual residual Ru and Rz in V ∗ , defined by Ru := b − a(uh, ·) and Rz := − a(·, zh), computable upper error bounds for e V ≤ ηu and f V ≤ ηz can be found by the arguments of the energy error estimators [1, 3]. Indeed, the parallelogram rule shows 2 (e) = 2 a(e, f) = e + f 2 a − e 2 a − f 2 a. This right-hand side can be written in terms of residuals, in the spirit of (3.1), namely e a = Ru V ∗ , f a = Rz V ∗ , and e + f a = Ru+z V ∗ for Ru+z := b + − a(uh + zh, ·) = Ru + Rz ∈ V ∗ . Therefore, the estimation of (e) is reduced to the computation of lower and upper error bounds for the three residuals Ru, Rz, and Ru+z with respect to the energy norm. This illustrates that the energy error estimation techniques of the previous sections may be employed for goal-oriented error control [1, 3]. Alternatively, Rannacher et al. investigated the weighted residual technique where the Clément interpolation operator is employed as well. The reader is re- ferred to [4].
- 54. 42 C. Carstensen References [1] Ainsworth, M. and Oden, J.T. (2000). A posteriori error estimation in finite element analysis. Wiley-Interscience [John Wiley Sons], New York. xx+240. [2] Babuška I. and Miller A. (1987) A feedback finite element method with a posteriori error estimation. I. The finite element method and some properties of the a posteriori estimator. Comp. Methods Appl. Mech. Engrg., 61, 1, 1–40. [3] Babuška, I. and Strouboulis, T. (2001). The finite element method and its reliability. The Clarendon Press Oxford University Press, New York, xii+802. [4] Bangerth, W. and Rannacher, R. (2003). Adaptive finite element methods for differ- ential equations. (Lectures in Mathematics ETH Zürich) Birkhäuser Verlag, Basel, viii+207. [5] Bank, R.E. and Weiser, A. (1985). Some a posteriori error estimators for elliptic partial differential equations. Math. Comp., 44, 170, 283–301. [6] Becker, R. and Rannacher, R. (1996). A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math, 4, 4, 237–264. [7] Becker, R. and Rannacher, R. (2001). An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica, Cambridge University Press, 2001, 1–102. [8] Binev, P., Dahmen, W. and DeVore, R. (2004). Adaptive finite element methods with convergence rates. Numer. Math., 97, 2, 219–268. [9] Carstensen, C. (1999). Quasi-interpolation and a posteriori error analysis in finite element method. M2AN Math. Model. Numer. Anal., 33, 6, 1187–1202. [10] Carstensen, C. (2004). Some remarks on the history and future of averaging tech- niques in a posteriori finite element error analysis. ZAMM Z. Angew. Math. Mech., 84, 1, 3–21. [11] Carstensen C. All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable. Math. Comp. 73 (2004) 1153-1165. [12] Carstensen, C. (2006). On the Convergence of Adaptive FEM for Convex Minimiza- tion Problems (in preparation). [13] Carstensen, C., Bartels, S. and Klose, R. (2001). An experimental survey of a pos- teriori Courant finite element error control for the Poisson equation. Adv. Comput. Math., 15, 1-4, 79–106. [14] Carstensen, C. and Funken, S.A. Constants in Clément-interpolation error and resid- ual based a posteriori estimates in finite element methods. East-West J. Numer. Math. 8(3) (2000) 153–175. [15] Carstensen, C. and Funken, S.A. Fully reliable localised error control in the FEM. SIAM J. Sci. Comp. 21(4) (2000) 1465–1484. [16] Carstensen, C. and Verfürth, R. (1999). Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal., 36, 5, 1571– 1587. [17] Ciarlet, P.G. The Finite Element Method for Elliptic Problems. North-Holland, 1978. (reprinted in the Classics in Applied Mathematics Series, SIAM, 2002)
- 55. Exploring the Variety of Random Documents with Different Content
- 56. and he indicated Bernard who seemed to me to have a greater confusion than the discovery gave a cause for. “Bernard has been good enough,” said I. “You discover two Scots, Father Hamilton, in a somewhat sentimental situation. The lady did me the honour to be interested in my little travels, and I did my best to keep her informed.” He turned away as he had been shot, hiding his face, but I saw from his neck that he had grown as white as parchment. “What in the world have I done?” thinks I, and concluded that he was angry for my taking the liberty to use the dismissed servant as a go-between. In a moment or two he turned about again, eying me closely, and at last he put his hand upon my shoulder as a schoolmaster might do upon a boy's. “My good Paul,” said he, “how old are you?” “Twenty-one come Martinmas,” I said. “Expiscate! elucidate! 'Come Martinmas,'” says he, “and what does that mean? But no matter—twenty-one says my barbarian; sure 'tis a right young age, a very baby of an age, an age in frocks if one that has it has lived the best of his life with sheep and bullocks.” “Sir,” I said, indignant, “I was in very honest company among the same sheep and bullocks.” “Hush!” said he, and put up his hand, eying me with compassion and kindness. “If thou only knew it, lad, thou art due me a civil attention at the very least. Sure there is no harm in my mentioning that thou art mighty ingenuous for thy years. 'Tis the quality I would be the last to find fault with, but sometimes it has its inconveniences. And Bernard”—he turned to the Swiss who was still greatly disturbed—“Bernard is a somewhat older gentleman. Perhaps he will say—our good Bernard—if he was the person I have to thank for taking the sting out of the wasp, for extracting the bullet from my pistol? Ah! I see he is the veritable person. Adorable Bernard, let that stand to his credit!” Then Bernard fell trembling like a saugh tree, and protested he did but what he was told.
- 57. “And a good thing, too,” said the priest, still very pale but with no displeasure. “And a good thing too, else poor Buhot, that I have seen an infinity of headachy dawns with, had been beyond any interest in cards or prisoners. For that I shall forgive you the rest that I can guess at. Take Monsieur Grog's letter where you have taken the rest, and be gone.” The Swiss went out much crestfallen from an interview that was beyond my comprehension. When he was gone Father Hamilton fell into a profound meditation, walking up and down his room muttering to himself. “Faith, I never had such a problem presented to me before,” said he, stopping his walk; “I know not whether to laugh or swear. I feel that I have been made a fool of, and yet nothing better could have happened. And so my Croque-mort, my good Monsieur Propriety, has been writing the lady? I should not wonder if he thought she loved him.” “Nothing so bold,” I cried. “You might without impropriety have seen every one of my letters, and seen in them no more than a seaman's log.” “A seaman's log!” said he, smiling faintly and rubbing his massive chin; “nothing would give the lady more delight, I am sure. A seaman's log! And I might have seen them without impropriety, might I? That I'll swear was what her ladyship took very good care to obviate. Come now, did she not caution thee against telling me of this correspondence?” I confessed it was so; that the lady naturally feared she might be made the subject of light talk, and I had promised that in that respect she should suffer nothing for her kindly interest in a countryman. The priest laughed consumedly at this. “Interest in her countryman!” said he. “Oh, lad, wilt be the death of me for thy unexpected spots of innocence.” “And as to that,” I said, “you must have had a sort of correspondence with her yourself.”
- 58. “I!” said he. “Comment!” “To be quite frank with you,” said I, “it has been the cause of some vexatious thoughts to me that the letter I carried to the Prince was directed in Miss Walkinshaw's hand of write, and as Buhot informed me, it was the same letter that was to wile his Royal Highness to his fate in the Rue des Reservoirs.” Father Hamilton groaned, as he did at any time the terrible affair was mentioned. “It is true, Paul, quite true,” said he, “but the letter was a forgery. I'll give the lady the credit to say she never had a hand in it.” “I am glad to hear that, for it removes some perplexities that have troubled me for a while back.” “Ah,” said he, “and your perplexities and mine are not over even now, poor Paul. This Bernard is like to be the ruin of me yet. For you, however, I have no fear, but it is another matter with the poor old fool from Dixmunde.” His voice broke, he displayed thus and otherwise so troubled a mind and so great a reluctance to let me know the cause of it that I thought it well to leave him for a while and let him recover his old manner. To that end I put on my coat and hat and went out rather earlier than usual for my evening walk.
- 59. I CHAPTER XXVIII THE MAN WITH THE TARTAN WAISTCOAT t was the first of May. But for Father Hamilton's birds, and some scanty signs of it in the small garden, the lengthened day and the kindlier air of the evenings, I might never have known what season it was out of the almanac, for all seasons were much the same, no doubt, in the Isle of the City where the priest and I sequestered. 'Twas ever the shade of the tenements there; the towers of the churches never greened nor budded; I would have waited long, in truth, for the scent of the lilac and the chatter of the rook among these melancholy temples. Till that night I had never ventured farther from the gloomy vicinity of the hospital than I thought I could safely retrace without the necessity of asking any one the way; but this night, more courageous, or perhaps more careless than usual, I crossed the bridge of Notre Dame and found myself in something like the Paris of the priest's rhapsodies and the same all thrilling with the passion of the summer. It was not flower nor tree, though these were not wanting, but the spirit in the air—young girls laughing in the by- going with merriest eyes, windows wide open letting out the sounds of songs, the pavements like a river with zesty life of Highland hills when the frosts above are broken and the overhanging boughs have been flattering it all the way in the valleys. I was fair infected. My step, that had been unco' dull and heavy, I fear, and going to the time of dirges on the Isle, went to a different tune; my being rhymed and sang. I had got the length of the Rue de Richelieu and humming to myself in the friendliest key, with the good-natured people pressing about me, when of a sudden it began to rain. There was no close in the neighbourhood where I could
- 60. shelter from the elements, but in front of me was the door of a tavern called the Tête du Duc de Burgoyne shining with invitation, and in I went. A fat wife sat at a counter; a pot-boy, with a cry of “V'ià!” that was like a sheep's complaining, served two ancient citizens in skull-caps that played the game of dominoes, and he came to me with my humble order of a litre of ordinary and a piece of bread for the good of the house. Outside the rain pelted, and the folks upon the pavement ran, and by-and-by the tavern-room filled up with shelterers like myself and kept the pot-boy busy. Among the last to enter was a group of five that took a seat at another corner of the room than that where I sat my lone at a little table. At first I scarcely noticed them until I heard a word of Scots. I think the man that used it spoke of “gully-knives,” but at least the phrase was the broadest lallands, and went about my heart. I put down my piece of bread and looked across the room in wonder to see that three of the men were gazing intently at myself. The fourth was hid by those in front of him; the fifth that had spoken had a tartan waistcoat and eyes that were like a gled's, though they were not on me. In spite of that, 'twas plain that of me he spoke, and that I was the object of some speculation among them. No one that has not been lonely in a foreign town, and hungered for communion with those that know his native tongue, can guess how much I longed for speech with this compatriot that in dress and eye and accent brought back the place of my nativity in one wild surge of memory. Every bawbee in my pocket would not have been too much to pay for such a privilege, but it might not be unless the overtures came from the persons in the corner. Very deliberately, though all in a commotion within, I ate my piece and drank my wine before the stare of the three men, and at last, on the whisper of one of them, another produced a box of dice.
- 61. “No, no!” said the man with the tartan waistcoat hurriedly, with a glance from the tail of his eye at me, but they persisted in their purpose and began to throw. My countryman in tartan got the last chance, of which he seemed reluctant to avail himself till the one unseen said: “Vous avez le de'', Kilbride.” Kilbride! the name was the call of whaups at home upon the moors! He laughed, shook, and tossed carelessly, and then the laugh was all with them, for whatever they had played for he had seemingly lost and the dice were now put by. He rose somewhat confused, looked dubiously across at me with a reddening face, and then came over with his hat in his hand. “Pardon, Monsieur,” he began; then checked the French, and said: “Have I a countryman here?” “It is like enough,” said I, with a bow and looking at his tartan. “I am from Scotland myself.” He smiled at that with a look of some relief and took a vacant chair on the other side of my small table. “I have come better speed with my impudence,” said he in the Hielan' accent, “than I expected or deserved. My name's Kilbride— MacKellar of Kilbride—and I am here with another Highland gentleman of the name of Grant and two or three French friends we picked up at the door of the play-house. Are you come off the Highlands, if I make take the liberty?” “My name is lowland,” said I, “and I hail from the shire of Renfrew.” “Ah,” said he, with a vanity that was laughable. “What a pity! I wish you had been Gaelic, but of course you cannot help it being otherwise, and indeed there are many estimable persons in the lowlands.” “And a great wheen of Highland gentlemen very glad to join them there too,” said I, resenting the implication.
- 62. “Of course, of course,” said he heartily. “There is no occasion for offence.” “Confound the offence, Mr. MacKellar!” said I. “Do you not think I am just too glad at this minute to hear a Scottish tongue and see a tartan waistcoat? Heilan' or Lowlan', we are all the same” when our feet are off the heather. “Not exactly,” he corrected, “but still and on we understand each other. You must be thinking it gey droll, sir, that a band of strangers in a common tavern would have the boldness to stare at you like my friends there, and toss a dice about you in front of your face, but that is the difference between us. If I had been in your place I would have thrown the jug across at them, but here I am not better nor the rest, because the dice fell to me, and I was one that must decide the wadger.” “Oh, and was I the object of a wadger?” said I, wondering what we were coming to. “Indeed, and that you were,” said he shamefacedly, “and I'm affronted to tell it. But when Grant saw you first he swore you were a countryman, and there was some difference of opinion.” “And what, may I ask, did Kilbride side with?” “Oh,” said he promptly, “I had never a doubt about that. I knew you were Scots, but what beat me was to say whether you were Hielan' or Lowlan'.” “And how, if it's a fair question, did you come to the conclusion that I was a countryman of any sort?” said I. He laughed softly, and “Man,” said he, “I could never make any mistake about that, whatever of it. There's many a bird that's like the woodcock, but the woodcock will aye be kennin' which is which, as the other man said. Thae bones were never built on bread and wine. It's a French coat you have there, and a cockit hat (by your leave), but to my view you were as plainly from Scotland as if you had a blue bonnet on your head and a sprig of heather in your lapels. And here am I giving you the strange cow's welcome (as the other man said), and that is all inquiry and no information. You must just be excusing our bit foolish wadger, and if the proposal would
- 63. come favourably from myself, that is of a notable family, though at present under a sort of cloud, as the other fellow said, I would be proud to have you share in the bottle of wine that was dependent upon Grant's impudent wadger. I can pass my word for my friends there that they are all gentry like ourselves—of the very best, in troth, though not over-nice in putting this task on myself.” I would have liked brawly to spend an hour out any company than my own, but the indulgence was manifestly one involving the danger of discovery; it was, as I told myself, the greatest folly to be sitting in a tavern at all, so MacKellar's manner immediately grew cold when he saw a swithering in my countenance. “Of course,” said he, reddening and rising, “of course, every gentleman has his own affairs, and I would be the last to make a song of it if you have any dubiety about my friends and me. I'll allow the thing looks very like a gambler's contrivance.” “No, no, Mr. MacKellar,” said I hurriedly, unwilling to let us part like that, “I'm swithering here just because I'm like yoursel' of it and under a cloud of my own.” “Dod! Is that so?” said he quite cheerfully again, and clapping down, “then I'm all the better pleased that the thing that made the roebuck swim the loch—and that's necessity—as the other man said, should have driven me over here to precognosce you. But when you say you are under a cloud, that is to make another way of it altogether, and I will not be asking you over, for there is a gentleman there among the five of us who might be making trouble of it.” “Have you a brother in Glasgow College?” says I suddenly, putting a question that had been in my mind ever since he had mentioned his name. “Indeed, and I have that,” said he quickly, “but now he is following the law in Edinburgh, where I am in the hopes it will be paying him better than ever it paid me that has lost two fine old castles and the best part of a parish by the same. You'll not be sitting there and telling me surely that you know my young brother Alasdair?”
- 64. “Man! him and me lodged together in Lucky Grant's, in Crombie's Land in the High Street, for two Sessions,” said I. “What!” said MacKellar. “And you'll be the lad that snow-balled the bylie, and your name will be Greig?” As he said it he bent to look under the table, then drew up suddenly with a startled face and a whisper of a whistle on his lips. “My goodness!” said he, in a cautious tone, “and that beats all. You'll be the lad that broke jyle with the priest that shot at Buhot, and there you are, you amadain, like a gull with your red brogues on you, crying 'come and catch me' in two languages. I'm telling you to keep thae feet of yours under this table till we're out of here, if it should be the morn's morning. No—that's too long, for by the morn's morning Buhot's men will be at the Hôtel Dieu, and the end of the story will be little talk and the sound of blows, as the other man said.” Every now and then as he spoke he would look over his shoulder with a quick glance at his friends—a very anxious man, but no more anxious than Paul Greig. “Mercy on us!” said I, “do you tell me you ken all that?” “I ken a lot more than that,” said he, “but that's the latest of my budget, and I'm giving it to you for the sake of the shoes and my brother Alasdair, that is a writer in Edinburgh. There's not two Scotchmen drinking a bowl in Paris town this night that does not ken your description, and it's kent by them at the other table there— where better?—but because you have that coat on you that was surely made for you when you were in better health, as the other man said, and because your long trams of legs and red shoes are under the table there's none of them suspects you. And now that I'm thinking of it, I would not go near the hospital place again.” “Oh! but the priest's there,” said I, “and it would never do for me to be leaving him there without a warning.” “A warning!” said MacKellar with contempt. “I'm astonished to hear you, Mr. Greig. The filthy brock that he is!”
- 65. “If you're one of the Prince's party,” said I, “and it has every look of it, or, indeed, whether you are or not, I'll allow you have some cause to blame Father Hamilton, but as for me, I'm bound to him because we have been in some troubles together.” “What's all this about 'bound to him'?” said MacKellar with a kind of sneer. “The dog that's tethered with a black pudding needs no pity, as the other man said, and I would leave this fellow to shift for himself.” “Thank you,” said I, “but I'll not be doing that.” “Well, well,” said he, “it's your business, and let me tell you that you're nothing but a fool to be tangled up with the creature. That's Kilbride's advice to you. Let me tell you this more of it, that they're not troubling themselves much about you at all now that you have given them the information.” “Information!” I said with a start. “What do you mean by that?” He prepared to join his friends, with a smile of some slyness, and gave me no satisfaction on the point. “You'll maybe ken best yourself,” said he, “and I'm thinking your name will have to be Robertson and yourself a decent Englishman for my friends on the other side of the room there. Between here and yonder I'll have to be making up a bonny lie or two that will put them off the scent of you.” A bonny lie or two seemed to serve the purpose, for their interest in me appeared to go no further, and by-and-by, when it was obvious that there would be no remission of the rain, they rose to go. The last that went out of the door turned on the threshold and looked at me with a smile of recognition and amusement. It was Buhot!
- 67. W CHAPTER XXIX WHEREIN THE PRIEST LEAVES ME, AND I MAKE AN INLAND VOYAGE hat this marvel betokened was altogether beyond my comprehension, but the five men were no sooner gone than I clapped on my hat and drew up the collar of my coat and ran like fury through the plashing streets for the place that was our temporary home. It must have been an intuition of the raised that guided me; my way was made without reflection on it, at pure hazard, and yet I landed through a multitude of winding and bewildering streets upon the Isle of the City and in front of the Hôtel Dieu in a much shorter time than it had taken me to get from there to the Duke of Burgundy's Head. I banged past the doorkeeper, jumped upstairs to the clergyman's quarters, threw open the door and—found Father Hamilton was gone! About the matter there could be no manner of dubiety, for he had left a letter directed to myself upon the drawers-head. “My Good Paul (said the epistle, that I have kept till now as a memorial of my adventure): When you return you will discover from this that I have taken leave a l'anglaise, and I fancy I can see my secretary looking like the arms of Bourges (though that is an unkind imputation). 'Tis fated, seemingly, that there shall be no rest for the sole of the foot of poor Father Hamilton. I had no sooner got to like a loose collar, and an unbuttoned vest, and the seclusion of a cell, than I must be plucked out; and now when my birds—the darlings! —are on the very point of hatching I must make adieux. Oh! la belle équipée! M. Buhot knows where I am—that's certain, so I must remove myself, and this time I do not propose to burden M. Paul
- 68. Greig with my company, for it will be a miracle if they fail to find me. As for my dear Croque-mort, he can have the glass coach and Jacques and Bernard, and doubtless the best he can do with them is to take all to Dunkerque and leave them there. I myself, I go sans trompette, and no inquiries will discover to him where I go.” As a postscript he added, “And 'twas only a sailor's log, dear lad! My poor young Paul!” When I read the letter I was puzzled tremendously, and at first I felt inclined to blame the priest for a scurvy flitting to rid himself of my society, but a little deliberation convinced me that no such ignoble consideration was at the bottom of his flight. If I read his epistle aright the step he took was in my own interest, though how it could be so there was no surmising. In any case he was gone; his friend in the hospital told me he had set out behind myself, and taken a candle with him and given a farewell visit to his birds, and almost cried about them and about myself, and then departed for good to conceal himself, in some other part of the city, probably, but exactly where his friend had no way of guessing. And it was a further evidence of the priest's good feeling to myself (if such were needed) that he had left a sum of a hundred livres for me towards the costs of my future movements. I left the Hôtel Dieu at midnight to wander very melancholy about the streets for a time, and finally came out upon the river's bank, where some small vessels hung at a wooden quay. I saw them in moonlight (for now the rain was gone), and there rose in me such a feeling as I had often experienced as a lad in another parish than the Mearns, to see the road that led from strangeness past my mother's door. The river seemed a pathway out of mystery and discontent to the open sea, and the open sea was the same that beat about the shores of Britain, and my thought took flight there and then to Britain, but stopped for a space, like a wearied bird, upon the town Dunkerque. There is one who reads this who will judge kindly, and pardon when I say that I felt a sort of tenderness for the lady there, who was not only my one friend in France, so far as I could guess, but, next to my mother, the only woman who knew my shame and still retained regard for me. And thinking about
- 69. Scotland and about Dunkerque, and seeing that watery highway to them both, I was seized with a great repugnance for the city I stood in, and felt that I must take my feet from there at once. Father Hamilton was lost to me: that was certain. I could no more have found him in this tanglement of streets and strange faces than I could have found a needle in a haystack, and I felt disinclined to make the trial. Nor was I prepared to avail myself of his offer of the coach and horses, for to go travelling again in them would be to court Bicêtre anew. There was a group of busses or barges at the quay, as I have said, all huddled together as it were animals seeking warmth, with their bows nuzzling each other, and on one of them there were preparations being made for her departure. A cargo of empty casks was piled up in her, lights were being hung up at her bow and stern, and one of her crew was ashore in the very act of casting off her ropes. At a flash it occurred to me that I had here the safest and the speediest means of flight. I ran at once to the edge of the quay and clumsily propounded a question as to where the barge was bound for. “Rouen or thereabouts,” said the master. I asked if I could have a passage, and chinked my money in my pocket. My French might have been but middling, but Lewis d'Or talks in a language all can understand. Ten minutes later we were in the fairway of the river running down through the city which, in that last look I was ever fated to have of it, seemed to brood on either hand of us like bordering hills, and at morning we were at a place by name Triel. Of all the rivers I have seen I must think the Seine the finest. It runs in loops like my native Forth, sometimes in great, wide stretches that have the semblance of moorland lochs. In that fine weather, with a sun that was most genial, the country round about us basked and smiled. We moved upon the fairest waters, by magic gardens, and the borders of enchanted little towns. Now it would be
- 70. a meadow sloping backward from the bank, where reeds were nodding, to the horizon; now an orchard standing upon grass that was the rarest green, then a village with rusty roofs and spires and the continual chime of bells, with women washing upon stones or men silent upon wherries fishing. Every link of the river opened up a fresher wonder; if not some poplared isle that had the invitation to a childish escapade, 'twould be another town, or the garden of a château, maybe, with ladies walking stately on the lawns, perhaps alone, perhaps with cavaliers about them as if they moved in some odd woodland minuet. I can mind of songs that came from open windows, sung in women's voices; of girls that stood drawing water and smiled on us as we passed, at home in our craft of fortune, and still the lucky roamers seeing the world so pleasantly without the trouble of moving a step from our galley fire. Sometimes in the middle of the days we would stop at a red- faced, ancient inn, with bowers whose tables almost had their feet dipped in the river, and there would eat a meal and linger on a pot of wine while our barge fell asleep at her tether and dreamt of the open sea. About us in these inns came the kind country-people and talked of trivial things for the mere sake of talking, because the weather was sweet and God so gracious; homely sounds would waft from the byres and from the barns—the laugh of bairns, the whistle of boys, the low of cattle. At night we moored wherever we might be, and once I mind of a place called Andelys, selvedged with chalky cliffs and lorded over by a castle called Gaillard, that had in every aspect of it something of the clash of weapons and of trumpet-cry. The sky shone blue through its gaping gables and its crumbling windows like so many eyes; the birds that wheeled all round it seemed to taunt it for its inability. The old wars over, the deep fosse silent, the strong men gone—and there at its foot the thriving town so loud with sounds of peaceful trade! Whoever has been young, and has the eye for what is beautiful and great and stately, must have felt in such a scene that craving for companionship that tickles like a laugh within the heart— that longing for some one to feel with him, and understand, and look
- 71. upon with silence. In my case 'twas two women I would have there with me just to look upon this Gaillard and the town below it. Then the bending, gliding river again, the willow and the aspen edges, the hazy orchards and the emerald swards; hamlets, towns, farm-steadings, châteaux, kirks, and mills; the flying mallard, the leaping perch, the silver dawns, the starry nights, the ripple of the water in my dreams, and at last the city of Rouen. My ship of fortune went no further on. I slept a night in an inn upon the quay, and early the next morning, having bought a pair of boots to save my red shoes, I took the road over a hill that left Rouen and all its steeples, reeking at the bottom of a bowl. I walked all day, through woods and meadows and trim small towns and orchards, and late in the gloaming came upon the port of Havre de Grace. The sea was sounding there, and the smell of it was like a salutation. I went out at night from my inn, and fairly joyed in its propinquity, and was so keen on it that I was at the quay before it was well daylight. The harbour was full of vessels. It was not long ere I got word of one that was in trim for Dunkerque, to which I took a passage, and by favour of congenial weather came upon the afternoon of the second day. Dunkerque was more busy with soldiers than ever, all the arms of France seemed to be collected there, and ships of war and flat- bottomed boats innumerable were in the harbour. At the first go-off I made for the lodgings I had parted from so unceremoniously on the morning of that noisy glass coach. The house, as I have said before, was over a baker's shop, and was reached by a common outer stair that rose from a court-yard behind. Though internally the domicile was well enough, indeed had a sort of old-fashioned gentility, and was kept by a woman whose man had been a colonel of dragoons, but now was a tippling pensioner upon the king, and his own wife's labours, it was, externally, somewhat mean, the place a solid merchant of our own country might inhabit, but scarce the place wherein to look for royal
- 72. blood. What was my astonishment, then, when, as I climbed the stair, I came face to face with the Prince! I felt the stair swing off below me and half distrusted my senses, but I had the presence of mind to take my hat off. “Bon jour, Monsieur, said he, with a slight hiccough, and I saw that he was flushed and meant to pass with an evasion. There and then a daft notion to explain myself and my relations with the priest who had planned his assassination came to me, and I stopped and spoke. “Your Royal Highness—-” I began, and at that he grew purple. “Cest un drôle de corps!” said he, and, always speaking in French, said he again: “You make an error, Monsieur; I have not the honour of Monsieur's acquaintance,” and looked at me with a bold eye and a disconcerting. “Greig,” I blurted, a perfect lout, and surely as blind as a mole that never saw his desire, “I had the honour to meet your Royal Highness at Versailles.” “My Royal Highness!” said he, this time in English. “I think Monsieur mistakes himself.” And then, when he saw how crestfallen I was, he smiled and hiccoughed again. “You are going to call on our good Clancarty,” said he. “In that case please tell him to translate to you the proverb, Oui phis sait plus se tait.” “There is no necessity, Monsieur,” I answered promptly. “Now that I look closer I see I was mistaken. The person I did you the honour to take you for was one in whose opinion (if he took the trouble to think of me at all) I should have liked to re-establish myself, that was all.” In spite of his dissipation there was something noble in his manner—a style of the shoulders and the hands, a poise of the head that I might practise for years and come no closer on than any nowt upon my father's fields. It was that which I remember best of our engagement on the stair, and that at the last of it he put out his hand to bid me good-day.
- 73. “My name,” says he, “is Monsieur Albany so long as I am in Dunkerque. À bon entendeur salut! I hope we may meet again, Monsieur Greig.” He looked down at the black boots I had bought me in Rouen. “If I might take the liberty to suggest it,” said he, smiling, “I should abide by the others. I have never seen their wearer wanting wit, esprit, and prudence—which are qualities that at this moment I desire above all in those that count themselves my friends.” And with that he was gone. I watched him descend the remainder of the stair with much deliberation, and did not move a step myself until the tip of his scabbard had gone round the corner of the close.
- 74. C CHAPTER XXX A GUID CONCEIT OF MYSELF LEADS ME FAR ASTRAY lancarty and Thurot were playing cards, so intent upon that recreation that I was in the middle of the floor before they realised who it was the servant had ushered in. “Mon Dieu! Monsieur Blanc-bec! Il n'y a pas de petit chez soi!” cried Thurot, dropping his hand, and they jumped to their feet to greet me. “I'll be hanged if you want assurance, child,” said Clancarty, surveying me from head to foot as if I were some curiosity. “Here's your exploits ringing about the world, and not wholly to your credit, and you must walk into the very place where they will find the smallest admiration.” “Not meaning the lodging of Captain Thurot,” said I. “Whatever my reputation may be with the world, I make bold to think he and you will believe me better than I may seem at the first glance.” “The first glance!” cried his lordship. “Gad, the first glance suggests that Bicêtre agreed with our Scotsman. Sure, they must have fed you on oatmeal. I'd give a hatful of louis d'or to see Father Hamilton, for if he throve so marvellously in the flesh as his secretary he must look like the side of St. Eloi. One obviously grows fat on regicide—fatter than a few poor devils I know do upon devotion to princes.” Thurot's face assured me that I was as welcome there as ever I had been. He chid Clancarty for his badinage, and told me he was certain all along that the first place I should make for after my flight
- 75. from Bicêtre (of which all the world knew) would be Dunkerque. “And a good thing too, M. Greig,” said he. “Not so good,” says I, “but what I must meet on your stair the very man-” “Stop!” he cried, and put his finger on his lip. “In these parts we know only a certain M. Albany, who is, my faith! a good friend of your own if you only knew it.” “I scarcely see how that can be,” said I. “If any man has a cause to dislike me it is his Roy—” “M. Albany,” corrected Thurot. “It is M. Albany, for whom, it seems, I was the decoy in a business that makes me sick to think on. I would expect no more than that he had gone out there to send the officers upon my heels, and for me to be sitting here may be simple suicide.” Clancarty laughed. “Tis the way of youth,” said he, “to attach far too much importance to itself. Take our word for it, M. Greig, all France is not scurrying round looking for the nephew of Andrew Greig. Faith, and I wonder at you, my dear Thurot, that has an Occasion here—a veritable Occasion—and never so much as says bottle. Stap me if I have a friend come to me from a dungeon without wishing him joy in a glass of burgundy!” The burgundy was forthcoming, and his lordship made the most of it, while Captain Thurot was at pains to assure me that my position was by no means so bad as I considered it. In truth, he said, the police had their own reasons for congratulating themselves on my going out of their way. They knew very well, as M. Albany did, that I had been the catspaw of the priest, who was himself no better than that same, and for that reason as likely to escape further molestation as I was myself. Thurot spoke with authority, and hinted that he had the word of M. Albany himself for what he said. I scarcely knew which pleased me best—that I should be free myself or that the priest should have a certain security in his concealment.
- 76. I told them of Buhot, and how oddly he had shown his complacence to his escaped prisoner in the tavern of the Duke of Burgundy's Head. At that they laughed. “Buhot!” cried his lordship. “My faith! Ned must have been tickled to see his escaped prisoner in such a cosy cachette as the Duke's Head, where he and I, and Andy Greig—ay! and this same priest— tossed many a glass, Ciel! the affair runs like a play. All it wants to make this the most delightful of farces is that you should have Father Hamilton outside the door to come in at a whistle. Art sure the fat old man is not in your waistcoat pocket? Anyhow, here's his good health....” === MISSING PAGES (274-288) ===
- 77. CHAPTER XXXI. THE BARD OF LOVE WHO WROTE WITH OLD MATERIALS
- 78. W CHAPTER XXXII. THE DUEL IN THE AUBERGE GARDEN hoever it was that moved at the instigation of Madame on my behalf, he put speed into the business, for the very next day I was told my sous-lieutenancy was waiting at the headquarters of the regiment. A severance that seemed almost impossible to me before I learned from the lady's own lips that her heart was elsewhere engaged was now a thing to long for eagerly, and I felt that the sooner I was out of Dunkerque and employed about something more important than the tying of my hair and the teasing of my heart with thinking, the better for myself. Teasing my heart, I say, because Miss Walkinshaw had her own reasons for refusing to see me any more, and do what I might I could never manage to come face to face with her. Perhaps on the whole it was as well, for what in the world I was to say to the lady, supposing I were privileged, it beats me now to fancy. Anyhow, the opportunity never came my way, though, for the few days that elapsed before I departed from Dunkerque, I spent hours in the Rue de la Boucherie sipping sirops on the terrace of the Café Coignet opposite her lodging, or at night on the old game of humming ancient love-songs to her high and distant window. All I got for my pains were brief and tantalising glimpses of her shadow on the curtains; an attenuate kind of bliss it must be owned, and yet counted by Master Red-Shoes (who suffered from nostalgia, not from love, if he had had the sense to know it) a very delirium of delight. One night there was an odd thing came to pass. But, first of all, I must tell that more than once of an evening, as I would be in the street and staring across at Miss Walkinshaw's windows, I saw his Royal Highness in the neighbourhood. His cloak might be
- 79. voluminous, his hat dragged down upon the very nose of him, but still the step was unmistakable. If there had been the smallest doubt of it, there came one evening when he passed me so close in the light of an oil lamp that I saw the very blotches on his countenance. What was more, he saw and recognised me, though he passed without any other sign than the flash of an eye and a halfstep of hesitation.
- 80. “H'm,” thinks I, “here's Monsieur Albany looking as if he might, like myself, be trying to content himself with the mere shadows of
- 81. things.” He saw me more than once, and at last there came a night when a fellow in drink came staving down the street on the side I was on and jostled me in the by-going without a word of apology. “Pardonnez, Monsieur!” said I in irony, with my hat off to give him a hint at his manners. He lurched a second time against me and put up his hand to catch my chin, as if I were a wench, “Mon Dieu! Monsieur Blanc-bec, 'tis time you were home,” said he in French, and stuttered some ribaldry that made me smack his face with an open hand. “I saw his Royal Highness in the neighbourhood—” At once he sobered with suspicious suddenness if I had had the sense to reflect upon it, and gave me his name and direction as one George Bonnat, of the Marine. “Monsieur will do me the honour of a meeting behind the Auberge Cassard after petit dejeuner to- morrow,” said he, and named a friend. It was the first time I was ever challenged. It should have rung in the skull of me like an alarm, but I cannot recall at this date that my heart beat a stroke the faster, or that the invitation vexed me more than if it had been one to the share of a bottle of wine. “It seems a pretty ceremony about a cursed impertinence on the part of a man in liquor,” I said, “but I'm ready to meet you either before or after petit déjeuner, as it best suits you, and my name's Greig, by your leave.” “Very well, Monsieur Greig,” said he; “except that you stupidly impede the pavement and talk French like a Spanish cow (comme une vache espagnole), you seem a gentleman of much accommodation. Eight o'clock then, behind the auberge,” and off went Sir Ruffler, singularly straight and business-like, with a profound congé for the unfortunate wretch he planned to thrust a spit through in the morning. I went home at once, to find Thurot and Clancarty at lansquenet. They were as elate at my story as if I had been asked to dine with Louis.
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