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“Book˙2011” — 2012/2/26 — 9:58 — page xv — #9
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Preface xv
The first two chapters, which deal with dyadic figures, are quite elemen-
tary. Except for very basic properties of Hausdorff measures, they should be
accessible to the beginning graduate students. The rest of the book presup-
poses the knowledge equivalent to the first year graduate course in analysis.
In addition, some familiarity with Hausdorff measures and distributions is ex-
pected. Rudimentary results from functional analysis are employed in the last
two chapters. Our presentation owes much to the excellent textbooks [29, 75]
and monographs [1, 33], which can serve as useful references.
During the preparation of this text I largely benefited from discussions
with L. Ambrosio, P. Bouafia, G.D. Chakerian, T. De Pauw, D.B. Fuchs,
N. Fusco, R.J. Gardner, G. Gruenhage, Z. Nashed, M. Šilhavý, S. Solecki,
V. Sverak, B.S. Thomson, and M. Torres. I am obliged to W.G. McCallum
who offered me a position of Research Associate in the Mathematics Depart-
ment of the University of Arizona; it gave me access to university facilities, in
particular to the university library.
Editorial help provided by the publisher was invaluable. In this regard my
thanks belong to K. Craig, M. Dimont, S. Kumar, S. Morkert, and R. Stern.
W.F.P.
Tucson, Arizona
February 2012
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-17-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 4 — #14
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4 1. Preliminaries
By a function we always mean an extended real-valued function. A finite
function is real-valued. If a function f equals identically to c ∈ R, we write
f ≡ c. When no confusion can arise, the same symbol denotes a function f
defined on a set A and its restriction f � B to a set B ⊂ A. For a function f
defined on a set E and t ∈ R, we let
{f > t} :=
�
x ∈ E : f(x) > t
�
,
and define the sets {f ≥ t}, {f < t}, etc. similarly. The set {f = 0} is called
the null set of f. Further, we let
f+
:= max{f, 0} and f−
:= max{−f, 0},
and note that f = f+
− f−
and |f| = f+
+ f−
. The value of f at x ∈ E is
denoted interchangeably by f(x), f[x], and �f, x�.
For m ∈ N, and x := (ξ1, . . . , ξm) and y := (η1, . . . , ηm) in Rm
,
x · y :=
m
�
i=1
ξiηi and |x| :=
√
x · x.
In Rm
we use exclusively the Euclidean metric induced by the norm |x|. The
diameter, closure, interior, and boundary of a set E ⊂ Rm
are denoted by
d(E), cl E, int E, and ∂E, respectively. The distance between sets A, B ⊂ Rm
is denoted by dist(A, B), or dist(x, B) if A = {x} is a singleton. Given E ⊂ Rm
and r ∈ R+, we let
U(E, r) : =
�
x ∈ Rm
: dist(x, E) < r
�
,
B(E, r) : =
�
x ∈ Rm
: dist(x, E) ≤ r
�
.
If E = {x} is a singleton, the sets
U(x, r) := U
�
{x}, r
�
and B(x, r) := B
�
{x}, r
�
are, respectively, the open and closed ball in Rm
of radius r centered at x. For
a pair of sets A, B ⊂ Rm
, the symbol A � B indicates that cl A is a compact
subset of int B.
Given E ⊂ Rm
and s ∈ N, we denote by C(E; Rs
) the linear space of
all continuous maps φ : E → Rs
. We let C(E) := C(E; R), and note that
according to this definition, all continuous function are real-valued.
The Urysohn function associated with a pair A, B of closed disjoint sub-
sets of Rm
is a function uA,B ∈ C(Rm
) defined by the formula
uA,B(x) :=
dist (x, B)
dist (x, A) + dist (x, B)
, x ∈ Rm
. (1.1.1)
Theorem 1.1.1 (Tietze). Let C ⊂ Rm
be a closed set. Each continuous map
φ : C → Rs
has a continuous extension ψ : Rm
→ Rs
.
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-21-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 6 — #16
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6 1. Preliminaries
Let E ⊂ Rm
and φ : E → Rs
. The indicator χE of E and the zero
extension φ of φ are defined by the formulae
χE(x) :=
�
1 if x ∈ E,
0 if x ∈ Rm
− E,
φ(x) :=
�
φ(x) if x ∈ E,
0 if x ∈ Rm
− E,
respectively. The support of φ is the set spt φ := cl {φ �= 0}.
LetΩ ⊂ Rm
be an open set. The linear space of all φ ∈ C(Ω; Rs
) with
spt φ � Ω is denoted by Cc(Ω; Rs
); the spaces Cc(Ω), Ck
c (Ω; Rs
), C∞
c (Ω; Rs
),
etc., are defined similarly. We always identify φ ∈ Cc(Ω; Rs
) with its zero
extension φ ∈ Cc(Rm
; Rs
). This simple convention, which will cause no con-
fusion, legitimizes the inclusions
Cc(U; Rs
) ⊂ Cc(Ω; Rs
) ⊂ Cc(Rm
; Rs
),
Ck
c (U; Rs
) ⊂ Ck
c (Ω; Rs
) ⊂ Ck
c (Rm
; Rs
)
(1.1.2)
where U ⊂ Ω is an open set and k = 1, 2, . . . , ∞.
Throughout this book, the ambient space is Rn
where n ≥ 1 is a fixed
integer. By {e1, . . . , en} we denote the standard base in Rn
, i.e.,
ei := (0, . . . , 1
i-th place
, . . . , 0), i = 1, . . . , n.
The projection in the direction of ei is the linear map
πi : x �→
�
j�=i
(x · ej)ej : Rn
→ Rn
.
As the setΠ i := πi(Rn
) is a linear subspace of Rn
with bases
e1, . . . , ei−1, ei+1, . . . , en,
it is isometric to Rn−1
. For each x ∈ Πi, the set π−1
i (x) is isometric to R.
Thus whenever convenient, we tacitly identify the spaceΠ i with Rn−1
, and
the set π−1
i (x) with R. Note that if n = 1, then π1(x) = 0 for each x ∈ R1
,
and hence π1(R1
) = {0}.
A cell in Rn
is the set A :=
�n
j=1[aj, bj] where aj < bj are real numbers.
If b1 − a1 = · · · = bn − an, the cell A is called a cube. A figure is a finite,
possibly empty, union of cells. A k-cube is a cube
n
�
j=1
�
ij2−k
, (ij + 1)2−k
�
where k and i1, . . . , in are integers. The family of all k-cubes is denoted by
Dk, and the elements of the union DC :=
�
k∈Z Dk are called dyadic cubes.
A dyadic figure is a finite, possibly empty, union of dyadic cubes. The family
of all dyadic figures in Rn
is denoted by DF.
At places we employ unspecified positive constants depending on certain
parameters, such as the dimension n. If κ is a constant depending only on
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-23-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 8 — #18
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8 1. Preliminaries
• absorbing if X =
�
{tC : t ∈ R},
• symmetric if −C = C,
• convex if tC + (1 − t)C ⊂ C for each 0 ≤ t ≤ 1.
The linear hull, or convex hull, of a set E ⊂ X is, respectively, the intersection
of all linear subspaces of X containing E, or the intersection of all convex
subsets of X containing E.
A topology in X for which the maps
(x, y) �→ x + y : X × X → X and (t, x) �→ tx : R × X → X
are continuous is called linear. Since each linear topology is induced by a
uniformity [28, Example 8.1.17], all topological linear spaces are completely
regular [28, Theorem 8.1.21]. A locally convex topology is a linear topology
that has a neighborhood base at zero consisting of convex sets. A linear space
equipped with a linear, or locally convex, topology is called a topological linear
space, or a locally convex space, respectively. In this book we encounter only
locally convex spaces.
A seminorm in X is a functional p : X → R such that
p(x + y) ≤ p(x) + p(y) and p(tx) = |t|p(x)
for all x, y ∈ X and each t ∈ R. Observe that p(0) = 0 ≤ p(x) for each x ∈ X.
A norm in X is a seminorm p such that p(x) = 0 implies x = 0. A family P
of seminorms is called separating if p(x) = 0 for all p ∈ P implies x = 0. A
separating family P defines a locally convex topology in X; the neighborhood
base at zero is given by convex symmetric sets
Up1,...,pk;ε :=
�
x ∈ X : max
�
p1(x), . . . , pk(x)
�
< ε
�
where p1, . . . , pk are in P and ε > 0. Conversely, each locally convex topology
in X is induced by a separating family P of seminorms [64, Remark 1.38, (b)].
The separating property of P guarantees that the topology defined by P is
Hausdorff. A locally convex topology induced by a countable separating family
{pk : k ∈ N} of seminorms is metrizable; for instance, by the metric
ρ(x, y) :=
∞
�
k=1
2−k pk(x − y)
1 + pk(x − y)
.
A Fréchet space is a completely metrizable locally convex space.
Even if a topology in X is defined by an uncountable family of seminorms,
there may exist another family of seminorms in X that is countable and defines
the same topology. The next example illustrates the situation.
Example 1.2.2. LetΩ ⊂ Rm
be an open set. The topology T of locally
uniform convergence in C(Ω; Rs
) is defined by the seminorms
pK(φ) := sup
x∈K
�
�φ(x)
�
�
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-25-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 9 — #19
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1.3. Measures 9
where K ⊂ Ω is a compact set and φ ∈ C(Ω; Rs
). Each x ∈ Ω ∩ Qm
has an
open neighborhood Ux � Ω. Organize {Ux : x ∈ Ω ∩ Qm
} into a sequence
U1, U2, . . . , and let Vj =
�j
i=1 Ui. Since Vj � Ω, and since each compact set
K ⊂ Ω is contained in some Vj, it is clear that T is defined by the seminorms
qj(φ) := sup
x∈Vj
�
�φ(x)
�
�, j = 1, 2 . . . .
It follows that T is the metrizable topology of uniform convergence on the
sets Vj. Thus T is complete, and
�
C(Ω; Rm
), T
�
is a Fréchet space.
Let (X, T) be a locally convex space. A set E ⊂ X is called bounded if for
each convex neighborhood U of zero there is t > 0 such that E ⊂ tU. Every
compact set E ⊂ X is bounded. In general, E ⊂ X is bounded if and only
if lim tkxk = 0 whenever {xk} is a sequence in E and {tk} is a sequence in
R converging to zero [64, Theorem 1.30]. If the topology T is induced by a
family P of seminorms, then E ⊂ X is bounded if and only if for each p ∈ P,
sup
�
p(x) : x ∈ E
�
< ∞.
The dual space of X, abreviated as the dual of X, is the linear space X∗
of all continuous linear functionals x∗
: X → R.1
To begin with, X∗
is just a
linear space with no topology. However, two locally convex topologies in X∗
are easy to introduce:
• the weak* topology W∗
defined by seminorms
x∗
�→
�
��x∗
, x�
�
� : X∗
→ R
where x ∈ X;
• the strong topology S∗
defined by seminorms
�x∗
�B := sup
��
��x∗
, x�
�
� : x ∈ B
�
where x∗
∈ X∗
and B ⊂ X is a bounded set.
Since each singleton {x} ⊂ X is a bounded set, the weak* topology is smaller
than the strong topology.
1.3. Measures
A measure 2
in an arbitrary set X is a function µ defined on all subsets of X
that satisfies the following conditions:
1A notable exception to the notation X∗ is the space D� of distributions defined in
Section 3.1 below.
2Our concept of measure is often called “outer measure”, and the term “measure” is
reserved for the restriction of “outer measure” to the family of all measurable sets. For our
purposes, such distinction is superfluous.
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-26-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 10 — #20
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10 1. Preliminaries
(i) µ(∅) = 0;
(ii) µ(B) ≤ µ(A) whenever B ⊂ A ⊂ X;
(iii) µ(
�∞
k=1 Ak) ≤
�∞
k=1 µ(Ak) whenever Ak ⊂ X for k = 1, 2, . . . .
Throughout this section, µ is a measure in a set X ⊂ Rm
. The reduction
of µ to a set Y ⊂ X is a measure µ Y in X defined by
(µ Y )(A) := µ(A ∩ Y )
for each A ⊂ X. If µ = µ Y , we say that µ lives in Y . A set E ⊂ X is
called µ measurable whenever
µ = µ E + µ (X − E).
The support of µ is the set
spt µ := X −
��
U ⊂ X : U is open in X and µ(U) = 0
�
.
Since each subset of Rm
has the Lindelöf property [28, Section 3.8 and Corol-
lary 4.1.16], we have µ(X − spt µ) = 0. In accordance with the standard
terminology, the measure µ is called
• σ-finite if X =
�∞
k=1 Ek and µ(Ek) < ∞ for k = 1, 2, . . . ,
• Borel if each relatively Borel subset of X is µ measurable,
• Borel regular if µ is a Borel measure and each E ⊂ X is contained in
a relatively Borel subset B of X such that µ(B) = µ(E),
• Radon if µ is a Borel regular measure and µ(K) < ∞ for each compact
set K ⊂ X,
• metric if µ(A ∪ B) ≥ µ(A) + µ(B) for each pair A, B ⊂ X such that
dist (A, B) > 0.
A set E ⊂ X is called µ σ-finite if the reduced measure µ E is σ-finite. The
next two theorems are proved in [29, Sections 1.1 and 1.9].
Theorem 1.3.1. Let Ω ⊂ Rm
be an open set. Each metric measure in Ω is
a Borel measure. If µ is a Borel regular measure in Ω, then µ E is a Radon
measure for each µ measurable set E ⊂ Ω with µ(E) < ∞. If µ is a Radon
measure in Ω, then the following conditions hold:
(1) For each set A ⊂ Ω,
µ(A) = inf
�
µ(U) : U ⊂ Ω is open and A ⊂ U
�
.
(2) For each µ measurable set A ⊂ Ω,
µ(A) = sup
�
µ(K) : K ⊂ A is compact
�
.
A set E ⊂ X with µ(E) = 0 is called µ negligible. Sets A, B ⊂ X are µ
equivalent if µ(A � B) = 0; they are µ overlapping if µ(A ∩ B) > 0. Maps φ
and ψ from a set E ⊂ X to a set Y are µ equivalent if the set
{φ �= ψ} :=
�
x ∈ E : φ(x) �= ψ(x)
�
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-27-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 11 — #21
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1.3. Measures 11
is µ negligible. When the measure µ is clearly understood from the context,
we indicate the equivalence by symbols A ∼ B and φ ∼ ψ.
Let E ⊂ X be a µ measurable set. A map φ : E → Rs
is called µ
measurable if the set φ−1
(B) is µ measurable for every Borel set B ⊂ Rs
. The
linear space of all µ measurable maps φ : E → Rs
is denoted by
L0
(E, µ; Rs
).
The essential support of φ ∈ L0
(E, µ; Rs
) is the set
ess spt φ := spt
�
µ {φ �= 0}
�
.
Unlike the support of φ, the essential support of φ depends only on the µ
equivalence class of φ. If φ ∈ L0
(E, µ; Rs
), we also define
ess sup
x∈E
�
�φ(x)
�
� := inf
�
sup
x∈E
�
�ψ(x)
�
� : ψ ∈ L0
(E, µ; Rs
) and ψ ∼ φ
�
.
For each φ ∈ L0
(E, µ; Rs
), there is ψ ∼ φ such that
ess spt φ = spt ψ and ess sup
x∈E
�
�φ(x)
�
� = sup
x∈E
�
�ψ(x)
�
�.
Convention 1.3.2. As is customary, we do not explicitly distinguish between
an individual set E ⊂ X, or an individual map φ, and the µ equivalence class
determined by E, or by φ, respectively. However, the reader should be aware
of the following custom: we think of the space L0
(E, µ; Rs
) as consisting of
equivalence classes, but when we write φ ∈ L0
(E, µ; Rs
), we view φ as a
specific representative of its equivalence class. In particular, when writing
φ ∈ L0
(E, µ; Rs
), we always assume that
ess spt φ = spt φ and ess sup
x∈E
�
�φ(x)
�
� = sup
x∈E
�
�φ(x)
�
�.
The next two theorems are standard tools of measure theory. Their proofs
can be found in [29, Section 1.2].
Theorem 1.3.3 (Egoroff). Let µ be a finite measure in X ⊂ Rm
, and let
{φk} be a sequence in L0
(X, µ; Rs
) that converges pointwise. Given ε > 0,
there is a µ measurable set E ⊂ X such that µ(X − E) < ε and the sequence
{φk � E} converges uniformly.
Theorem 1.3.4 (Luzin). Let µ be a finite Borel regular measure in X ⊂ Rm
,
and let φ ∈ L0
(X, µ; Rs
). Given ε > 0, there is a compact set K ⊂ X such
that µ(X − K) < ε and the restriction φ � K is continuous.
Given a µ measurable set E ⊂ X, we let
�φ�Lp(E,µ;Rs) : =
��
E
|φ|p
dµ
�1/p
if 1 ≤ p < ∞,
�φ�L∞(E,µ;Rs) : = ess sup
x∈E
�
�φ(x)
�
�
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-28-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 12 — #22
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12 1. Preliminaries
for each φ ∈ L0
(E, µ; Rs
), and for 1 ≤ p ≤ ∞, define
Lp
(E, µ; Rs
) : =
�
f ∈ L0
(E, µ; Rs
) : �f�Lp(E,µ;Rs) < ∞
�
.
Let 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞ be such that 1/p + 1/q = 1 where we define
1/∞ := 0. The Hölder inequality
�fg�L1(E,µ;Rs) ≤ �f�Lp(E,µ;Rs)�g�Lq(E,µ;Rs) (1.3.1)
holds for each f, g ∈ L0
(E, µ; Rs
); see [63, Theorem 3.5].
Let µ be a Borel measure in an open setΩ ⊂ Rm
. For 1 ≤ p ≤ ∞, we
denote by Lp
loc(Ω, µ; Rs
) the linear space of all maps φ ∈ L0
(Ω, µ; Rs
) such
that φ � U belongs to Lp
(U, µ; Rs
) for each open set U � Ω. Unless stated
otherwise, throughout we assume that Lp
loc(Ω, µ; Rs
) has been equipped with
the Fréchet topology defined by seminorms
φ �→ �φ � U�Lp(U,µ;Rs) : Lp
loc(Ω, µ; Rs
) → R
where U � Ω is an open set; cf. Example 1.2.2. We write Lp
(E, µ) and
Lp
loc(Ω, µ) instead of Lp
(E, µ; R) and Lp
loc(Ω, µ; R), respectively.
The following theorem is essential for establishing Theorem 2.3.7 and
Proposition 7.4.3 below; in addition, it simplifies proofs of some differentiation
results (Theorems 4.3.4 and 6.2.3 below). We call it the Henstock lemma, but
the name Saks-Henstock lemma is also used — cf. [44] and [37].
Theorem 1.3.5. Let µ be a Radon measure in X ⊂ Rm
, let E ⊂ X be a µ
measurable set with µ(E) < ∞, and let f ∈ L1
(E, µ) be real-valued. Given
ε > 0, there is δ : E → R+ satisfying the following condition: for every
collection {E1, . . . , Ep} of µ measurable µ nonoverlapping subsets of E, and
for every set of points {x1, . . . , xp} ⊂ E, the inequality
p
�
i=1
�
�
�
�f(xi)µ(Ei) −
�
Ei
f dµ
�
�
�
� < ε
holds whenever d
�
Ei ∪ {xi}
�
< δ(xi) for i = 1, . . . , p.
Proof. Choose ε > 0, and using the Vitali-Carathéodory theorem [63, The-
orem 2.25], find functions g and h defined on E that are, respectively, upper
and lower semicontinuous, and satisfy
g ≤ f ≤ h and
�
E
(h − g) dµ < ε.
There is δ : E → R+ such that g(y) < f(x) + ε and h(y) > f(x) − ε for all
x, y ∈ E with |x − y| < δ(x). If {E1, . . . , Ep} and {x1, . . . , xp} satisfy the
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-29-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 13 — #23
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1.4. Hausdorff measures 13
conditions of the proposition, then
�
Ei
g dµ ≤
�
Ei
f dµ ≤
�
Ei
h dµ,
�
Ei
g dµ − εµ(Ei) ≤ f(xi)µ(Ei) ≤
�
Ei
h dµ + εµ(Ei)
for i = 1, . . . , p. Consequently
p
�
i=1
�
�
�
�f(xi)µ(Ei) −
�
Ei
f dµ
�
�
�
� ≤
p
�
i=1
��
Ei
(h − g) dµ + εµ(Ei)
�
≤
�
E
(h − g) dµ + εµ(E)
< ε
�
1 + µ(E)
�
.
Lebesgue measure in Rm
is denoted by Lm
. For each subset E of the
ambient space Rn
, we let
|E| := Ln
(E).
Sets A, B ⊂ Rn
are called overlapping if they are Ln
overlapping, that is to
say if |A ∩ B| > 0. Unless specified otherwise, all concepts connected with
measures, such as “measurable”, “negligible”, etc., as well as the expressions
“almost all” and “almost everywhere”, refer to the measure Ln
in Rn
. For a
measurable set E ⊂ Rn
, we let
Lp
(E; Rs
) := Lp
(E, Ln
; Rs
) and Lp
(E) := Lp
(E, Ln
).
IfΩ ⊂ Rn
is an open set, the meanings of Lp
loc(Ω; Rs
) and Lp
loc(Ω) are obvious.
When no confusion is possible, we write
�
E
f(x) dx or
�
E
f instead of
�
E
f dLn
.
1.4. Hausdorff measures
We define Hausdorff measures in Rn
, and state some of their elementary
properties. Select a fixed s ≥ 0, and let
Γ(s) :=
� ∞
0
ts−1
e−t
dt and α(s) :=
Γ
�1
2
�s
Γ
�s
2 + 1
�.
Recall that Γ : t �→ Γ(t) is the classical Euler’s gamma function [62, Defini-
tion 8.17]. Using Fubini’s theorem and induction, we obtain
α(n) = Ln
��
x ∈ Rn
: |x| ≤ 1
��
;
a more advanced calculation can be found in [56, Chapter 1, Equation 1.1.7].
The function α : t �→ α(t) maps [0, ∞) to [1, 5), has only one local maximum
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-30-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 14 — #24
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14 1. Preliminaries
and one local minimum, attained at 5 < tmax < 6 and tmin = 0, respectively;
in addition α(t) → 0 as t → ∞.
For E ⊂ Rn
and δ > 0, let
Hs
δ(E) := inf
∞
�
k=1
α(s)
�
d(Ck)
2
�s
(1.4.1)
where the infimum is taken over all sequences {Ck} of subsets of Rn
such that
E ⊂
�∞
k=1 Ck and d(Ck) < δ for k = 1, 2, . . . ; here we define 00
:= 1 and
d(∅)s
:= 0. Letting
Hs
(E) := sup
δ>0
Hs
δ(E) = lim
δ→0
Hs
δ(E),
the function Hs
: E �→ Hs
(E), defined for each E ⊂ Rn
, is a measure in Rn
,
called the s-dimensional Hausdorff measure. Since
Hs
(A ∪ B) = Hs
(A) + Hs
(B)
for every pair of sets A, B ⊂ Rn
with dist(A, B) > 0, it follows from The-
orem 1.3.1 that Hs
is a Borel measure in Rn
. In addition, the measure Hs
is Borel regular by Proposition 1.4.2 below. However, Hs
is not a Radon
measure in Rn
when s < n.
It is easy to verify that H0
is the counting measure in Rn
. The constant
α(s)/2s
in the definition of Hs
δ(E) implies Hn
= Ln
. This equality follows
(nontrivially) from the isodiametric inequality
Ln
(E) ≤ α(n)
�
d(E)
2
�n
(1.4.2)
which holds for every E ⊂ Rn
; see [29, Section 2.2]. As the diameters of sets
are invariant with respect to isometric transformations, so are the Hausdorff
measures. Moreover, for each E ⊂ Rn
and every t > 0,
Hs
(tE) = ts
Hs
(E).
Remark 1.4.1. The value of Hs
δ(E), and a fortiori that of Hs
(E), does not
change when the sequence {Ck} in the defining equality (1.4.1) is assumed to
have one of the following additional properties:
(1) Each Ck is convex; since the diameters of Ck and its convex hull are
the same.
(2) Each Ck is closed; since the diameters of Ck and its closure cl Ck are
the same.
(3) Each Ck is open; since given ε > 0, we can find rk > 0 such that
d
�
U(Ck, rk)
�
< δ and d
�
U(Ck, rk)
�s
< d(Ck)s
+ ε2−k
for k = 1, 2, . . . .
(4) Each Ck is contained in E; since E is covered by the family {Ck ∩ E}
and d(Ck ∩ E) ≤ d(Ck) for k = 1, 2, . . . .
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-31-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 16 — #26
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16 1. Preliminaries
Proposition 1.4.4. Let s ≥ 0 and E ⊂ Rn
. Given ε > 0 and p ∈ Z, there is
a family Q ⊂
�
k≥p Dk such that E meets each Q ∈ Q, E ⊂ int
��
Q
�
, and
�
Q∈Q
d(Q)s
≤ β
�
Hs
(E) + ε
�
where β = β(n) > 0.
Proof. If s > n, the proposition holds with β = 1; the proof is analogous to
that of the second part of Proposition 1.4.3. Hence assume s ≤ n, and choose
ε > 0 and p ∈ Z. There is a cover {Cj} of E such that the diameter of each
Cj is smaller than δ = 2−p
, and
∞
�
j=1
α(s)
�
d(Cj)
2
�s
≤ Hs
δ(E) + ε ≤ Hs
(E) + ε.
Find an integer pj ≥ p with 2−pj −1
≤ d(Cj) < 2−pj
, and note
d(Cj) < d(Q)/
√
n = 2−pj
≤ 2d(Cj)
for every pj-cube Q. Select a pj-cube Q with Q ∩ Cj �= ∅, and denote by
Q1,j, . . . , Q3n,j all pj-cubes which meet Q, including Q itself. It follows that
Cj ⊂ int
��3n
i=1 Qi,j
�
, and hence E ⊂ int
��∞
j=1
�3n
i=1 Qi,j
�
. Moreover,
∞
�
j=1
3n
�
i=1
d(Qi,j)s
≤ 3n
·
�
2
√
n
�s
∞
�
j=1
d(Cj)s
≤ 3n (4n)s
α(s)
�
Hs
(E) + ε
�
.
Since 0 ≤ s ≤ n implies α(s) ≥ min
�
1, α(n)
�
, the desired inequality holds
with β := (12n)n
max
�
1, 1/α(n)
�
. Finally, replacing Q by a smaller family
{Q ∈ Q : Q ∩ E �= ∅} completes the proof.
Additional properties of Hausdorff measures in Rn
can be found in [30]
and [46]. Hausdorff measures defined in general metric spaces are investigated
in [59].
1.5. Differentiable and Lipschitz maps
LetΩ ⊂ Rn
be an open set. A map φ : Ω → Rm
is differentiable at x ∈ Ω if
there is a linear map L : Rn
→ Rm
such that
lim
y→x
�
�φ(y) − φ(x) − L(y − x)
�
�
|y − x|
= 0.
If such a map L exists, it is unique. We call it the derivative of φ at x, denoted
by Dφ(x). If a map φ = (f1, . . . , fn) from Ω to Rn
is differentiable at x ∈ Ω,
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-33-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 18 — #28
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18 1. Preliminaries
Proof. By [29, Section 3.1.1, Theorem 1], there is θ ∈ Lip(Rn
; Rm
) such that
θ(x) = φ(x) for each x ∈ E and Lip θ ≤
√
m Lip φ. As there is nothing to prove
otherwise, assume c := �φ�L∞(E;Rm) belongs to R+. Define γ : Rm
→ Rm
by
γ(y) :=
�
c y
|y| if |y| > c,
y if |y| ≤ c.
Since Lip γ = 1 and |γ|L∞(Rm;Rm) ≤ c, the composition ψ := γ ◦ θ is the
desired extension of φ.
With a considerable effort, one can improve on Proposition 1.5.2 by show-
ing that Lip ψ = Lip φ. This stronger result is called Kirschbraun’s theorem
[33, Theorem 2.10.43].
The next well-known theorem has several proofs of various levels of so-
phistication, e.g., [33, Theorem 3.1.6], [1, Theorem 2.14], or [29, Section 6.2,
Theorem 2]. For a proof with minimal prerequisites we refer to [29, Sec-
tion 3.1.2].
Theorem 1.5.3 (Rademacher). Each φ ∈ Lip(Rn
; Rm
) is differentiable at
almost all x ∈ Rn
.
Let E ⊂ Rn
and 0 ≤ s ≤ 1. The s-Hölder constant at x ∈ E of a map
φ : E → Rm
is the extended real number
Hsφ(x) := lim sup
y→x
y∈E
�
�φ(y) − φ(x)
�
�
|y − x|s
.
Clearly, H0φ(x) < ∞ if and only if φ is bounded in a neighborhood of x, and
H0φ(x) = 0 if and only if φ is continuous at x. If Hsφ(x) < ∞ and 0 ≤ t < s,
then Htφ(x) = 0. We call
Lip φ(x) := H1φ(x)
the Lipschitz constant of φ at x, and say that φ is Lipschitz at x whenever
Lip φ(x) < ∞. We say that φ is pointwise Lipschitz in a set C ⊂ E if it
is Lipschitz at each x ∈ C. A pointwise Lipschitz map in C need not be
Lipschitz in C, even if C is compact [51, Section 1.6].
Theorem 1.5.4 (Stepanoff). Let Ω ⊂ Rn
be an open set, and assume that
φ : Ω → Rm
is pointwise Lipschitz in a set E ⊂ Ω. Then φ is differentiable
at almost all x ∈ E.
For a proof of this generalization of Rademacher’s theorem we refer to
[33, Theorem 3.1.9], or to Section 6.2 below where a slightly more general
theorem is proved in detail; see Remark 7.2.4.
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-35-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 19 — #29
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1.5. Differentiable and Lipschitz maps 19
Theorem 1.5.5 (Whitney). Let K ⊂ Rn
be a compact set, and let f ∈ C(K)
and v ∈ C(K; Rn
) satisfy the following condition: given ε > 0, we can find
δ > 0 so that �
�f(y) − f(x) − v(x) · (y − x)
�
� ≤ ε|y − x|
for all x, y ∈ K with |y − x| < δ. There is g ∈ C1
(Rn
) such that
g(x) = f(x) and Dg(x) = v(x)
for each x ∈ K.
Theorem 1.5.5 is a special case of Whitney’s extension theorem. Proofs
of the general Whitney’s result, which implies the special case, can be found
in [29, Section 6.5] or in [70, Chapter 6, Section 2].
Let φ = (f1, . . . , fn) be a Lipschitz map from Rn
to Rn
. Then
Df1
. . .
Dfn
is an n × n matrix, whose determinant is denoted by det Dφ. The Jaco-
bian of φ is the function Jφ = | det Dφ| defined almost everywhere in Rn
by
Rademacher’s theorem. In view of (1.5.1), the inequality
�Jφ�L∞(Rn) ≤ (Lip φ)n
(1.5.2)
is a consequence of [29, Section 3.3, Lemma 1].
The next result is called interchangeably the area theorem or change of
variables theorem. It follows from [29, Section 3.3, Theorem 2].
Theorem 1.5.6. Let φ : Rn
→ Rn
be a Lipschitz map. If g ∈ L0
(Rn
) and
g ≥ 0, then y �→
��
g(x) : x ∈ φ−1
(y)} is a measurable function on Rn
and
�
Rn
g(x)Jφ(x) dx =
�
Rn
�
x∈φ−1(y)
g(x) dy. (1.5.3)
Employing Hausdorff measures and more elaborete Jacobians, formulas
similar to (1.5.3) hold for Lipschitz maps φ : Rn
→ Rm
where m �= n. On a few
occasions when such formulas are used, we refer the reader to the appropriate
sections of [29, Chapter 3].
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-36-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 22 — #32
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22 2. Divergence theorem for dyadic figures
Proposition 2.1.2. Let A be a cell, and let v ∈ C(A; Rn
) be differentiable at
each x ∈ int A. If div v belongs to L1
(A), then
�
A
div v(x) dx =
�
∂A
v · νA dHn−1
.
Proof. Let v = (v1, . . . , vn) and A =
�n
i=1[a−
i , a+
i ]. If
A±
i := {a±
i } × πi(int A),
then νA(x) = ±ei whenever x ∈ A±
i , i = 1, . . . , n. The boundary ∂A differs
from
�n
i=1(A−
i ∪ A+
i ) by an Hn−1
negligible set. Fix i and for x ∈ int A,
write x = (u, t) where u = πi(x) and t = x · ei. By Fubini’s theorem and the
fundamental theorem of calculus,
�
A
Divi(x) dx =
�
πi(int A)
�� a+
i
a−
i
d
dt
vi(u, t) dt
�
du
=
�
πi(int A)
�
vi(u, a+
i ) − vi(u, a−
i )
�
du
=
�
A+
i
v · νA dHn−1
+
�
A−
i
v · νA dHn−1
.
Summing up these equalities over i = 1, . . . , n completes the proof.
Corollary 2.1.3. Let E ⊂ Rn
, x ∈ E, and let {Ck} be a sequence of cubes
such that lim d(Ck) = 0. Assume that Ck ⊂ E and x ∈ Ck for k = 1, 2, . . . ,
and that v : E → Rm
belongs to L1
(∂C, Hn−1
; Rn
) for each cube C ⊂ E.
(1) If 0 ≤ s ≤ 1, then
lim sup
1
d(Ck)n−1+s
�
∂Ck
v · νCk
dHn−1
≤ 2nHsv(x).
(2) If x ∈ int E and v is differentiable at x, then
lim
1
|Ck|
�
∂Ck
v · νCk
dHn−1
= div v(x).
Proof. Choose ε > 0 — you can never go wrong by doing so. We may
assume Hsv(x) < ∞, and find δ > 0 so that
�
�v(y) − v(x)
�
� ≤
�
Hsv(x) + ε
�
· |y − x|s
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-39-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 24 — #34
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24 2. Divergence theorem for dyadic figures
see [29, Section 3.3.4, B]. Let vn := v · en and νn := νB · en. Observe
νn ◦ φ± = (νB ◦ φ±) · en =
φ± · en
r
= ±
g
r
.
If Uk = πn
�
U(0, 1 − 2−k)
�
, then the maps φ± � Uk are Lipschitz and U =
�∞
k=1 Uk. Thus
applying [29, Section 3.3.4, B] to each Uk, we obtain
�
(∂B)±
vnνn dHn−1
=
�
U
�
(vnνn) ◦ φ±
�
J dLn
= ±
�
U
vn
�
u, ±g(u)
�
du.
Now Fubini’s theorem and the fundamental theorem of calculus imply
�
B
Dnvn(x) dx =
�
U
�� g(u)
−g(u)
d
dt
vn(u, t) dt
�
du
=
�
U
vn
�
u, g(u)
�
du −
�
U
vn
�
u, −g(u)
�
du
=
�
(∂B)+
vnνn dHn−1
+
�
(∂B)−
vnνn dHn−1
=
�
∂B
vnνn dHn−1
,
since the boundary ∂B differs from (∂B)+ ∪ (∂B)− by an Hn−1 negligible set. The propo-
sition follows from symmetry.
2.2. Dyadic partitions
A partition is a finite (possibly empty) collection
P :=
�
(E1, x1), . . . , (Ep, xp)
�
where {E1, . . . , Ep} is a collection of nonoverlapping subsets of Rn
such that
xi ∈ Ei for i = 1, . . . , p. The body of P is the union [P] :=
�p
i=1 Ei, and
P is called a partition in a set A ⊂ Rn
if [P] ⊂ A. Given a set E ⊂ Rn
and δ : E → R+, we say that P is δ-fine if xi ∈ E and d(Ei) < δ(xi) for
i = 1, . . . , p. When each set Ei is a dyadic cube, then P is called a dyadic
partition.
If dyadic cubes A and B overlap, then either A ⊂ B or B ⊂ A. Conse-
quently, every family C of dyadic cubes has a nonoverlapping subfamily Q such
that
�
Q =
�
C. Dyadic cubes A and B are called adjacent if d(A) = d(B)
and A ∩ B �= ∅. Every dyadic cube is adjacent to 3n
dyadic cubes, including
itself. Recall that for an integer k the family of all k-cubes is denoted by Dk.
Given a family E of subsets of Rn
and x ∈ Rn
, we let
St(x, E) := {E ∈ E : x ∈ E}.
For each x ∈ Rn
and each k ∈ Z, the collection St(x, Dk) consists of at most
2n
k-cubes, and x belongs to the interior of
�
St(x, Dk). A star cover of
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-41-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 26 — #36
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26 2. Divergence theorem for dyadic figures
Now Ck ⊂ D≥k and Ek ⊂ Bk. Since each C ∈ Ck meets Ek, the definition of
Bk implies C ∈ B. Thus Ck ⊂ B, and we see that the family C :=
�∞
k=1 Ck
is a star cover of E that satisfies condition (1). Letting κ := 3n
β, we obtain
�
C∈C
d(C)t
≤
∞
�
k=1
�
C∈Ck
d(C)t
≤ κ
∞
�
k=1
�
Ht
(Ek) + ε2−k
�
= κ
�
Ht
(E) + ε
�
.
Proposition 2.2.2. Let E be a family of disjoint subsets of a dyadic figure A,
and for each E ∈ E select real numbers 0 ≤ tE ≤ n and εE > 0. Given
δ : A → R+, there is a δ-fine dyadic partition
P :=
�
(C1, x1), . . . , (Cp, xp)
�
such that [P] = A, and with a fixed κ = κ(n) > 0, the inequality
�
xi∈E
d(Ci)tE
≤ κ
�
HtE
(E) + εE
�
holds for each E ∈ E.
Proof. There is k ∈ N such that A is the union of k-cubes. Enlarging E
and making δ smaller, we may assume
�
E = A and δ(x) < 2−k
√
n for each
x ∈ A. Let CE be a star cover of E ∈ E associated with δE := δ � E, tE, and
εE according to Lemma 2.2.1. For every C ∈ CE, select xC ∈ E ∩ C with
d(C) < δE(xC). Since C :=
�
E∈E CE is a star cover of the compact set A,
there are nonoverlapping cubes C1, . . . , Cp in C such that A ⊂
�p
i=1 Ci. It
follows that
P :=
�
(Ci, xCi
) : |Ci ∩ A| > 0
�
is a δ-fine dyadic partition with A ⊂ [P]. As our assumption about δ implies
Ci ⊂ A whenever |Ci ∩ A| > 0, we obtain [P] = A. Since E is a disjoint
family, {Ci : xCi
∈ E} ⊂ CE for each E ∈ E. Hence with the same κ as in
Lemma 2.2.1, the inequality
�
xCi
∈E
d(Ci)tE
≤
�
C∈CE
d(C)tE
≤ κ
�
HtE
(E) + εE
�
holds for every E ∈ E.
Remark 2.2.3. Lemma 2.2.1 and Proposition 2.2.2 are due to W.B. Jurkat
[42, Section 4]. The classical Cousin’s lemma [17] or [51, Lemma 2.6.1], as well
as its generalization obtained by E.J. Howard [41, Lemma 5], are immediate
consequences of Proposition 2.2.2.
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-43-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 27 — #37
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2.3. Admissible maps 27
2.3. Admissible maps
Definition 2.3.1. Let E ⊂ Rn
be any set. A map φ : E → Rm
is called
admissible if there are numbers 0 ≤ sk < 1, and disjoint, possibly empty, sets
Ek ⊂ E such that φ is pointwise Lipschitz in E−
�∞
k=1 Ek and for k = 1, 2, . . . ,
the following conditions hold:
(i) Ek is Hn−1+sk
σ-finite, and Hsk
φ(x) < ∞ for each x ∈ Ek;
(ii) Hn−1+sk
(Ek) > 0 implies Hsk
φ(x) = 0 for each x ∈ Ek.
The family of all admissible maps from the set E to Rm
is denoted by
Adm(E; Rm
), and we write Adm(E) instead of Adm(E; R).
Note that in Definition 2.3.1 no topological restrictions are placed on the
exceptional sets Ek.
Remark 2.3.2. If Hn−1+sk
(Ek) = ∞, then Ek =
�
j∈N Ek,j where Ek,j are
disjoint sets with Hn−1+sk
(Ek,j) < ∞. Thus replacing each pair (Ek, sk)
with Hn−1+sk
(Ek) = ∞ by the collection
�
(Ek,j, sk) : j ∈ N
�
, condition (i)
of Definition 2.3.1 can be replaced by the condition:
(i*) Hn−1+sk
(Ek) < ∞, and Hsk
φ(x) < ∞ for each x ∈ Ek.
This observation will simplify future arguments.
Remark 2.3.3. Let E ⊂ Rn
and φ ∈ Adm(E; Rm
). Then H0φ(x) < ∞ for
all x ∈ E, and H0φ(x) = 0 for all x ∈ E − T where T ⊂ E is Hn−1
negligible.
Thus φ is locally bounded in E and continuous in E −T. It follows that if E is
Hn−1
measurable then so is φ, and if E is compact then φ is bounded. Since
each set Ek is negligible, Stepanoff’s theorem implies that φ is differentiable
at almost all x ∈ int E. The restriction φ � B belongs to Adm(B; Rm
) for
each B ⊂ A.
Remark 2.3.4. A commonly encountered map φ : E → Rm
is locally
bounded in E, continuous outside an Hn−1
negligible set T ⊂ E, and point-
wise Lipschitz outside an Hn−1
σ-finite set S ⊂ E [53, Theorem 2.9]. Letting
sk := 0 for k = 1, 2, . . . ,
E1 := T, E2 := S − T, and Ek = ∅
for k = 3, 4, . . . , we see that φ is admissible. Since locally bounded and
pointwise Lipschitz are extreme points of the scale represented by Hölder
constants, considering admissible maps is natural.
Proposition 2.3.5. Let E ⊂ Rn
. With respect to pointwise addition and
multiplication, Adm(E) is a commutative ring, and Adm(A; Rm
) is a module
over Adm(E).
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-44-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 28 — #38
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28 2. Divergence theorem for dyadic figures
Proof. Choosing φ,ψ ∈ Adm(E; Rm
) and g ∈ Adm(E), it suffices to show
that φ + ψ and gφ belong to Adm(E; Rm
). As the other proof is similar,
we show only that θ = gφ belongs to Adm(E; Rm
). Let {rk} and {sk} be
sequences in [0, 1), and {Ak} and {Bk} be sequences of disjoint subsets of E,
associated with φ and g, respectively, according to Definition 2.3.1. Further
let A0 := E −
�
k∈N Ak, B0 := E −
�
k∈N Bk, and r0 = s0 = 1. Define
ti,j := min{ri, sj} and Ei,j := Ai ∩ Bj
for i, j = 0, 1, . . . , and observe that 0 ≤ ti,j < 1 whenever (i, j) �= (0, 0),
and that E is the union of a disjoint collection {Ei,j : i, j = 0, 1, . . . }. By
Remark 2.3.3, both φ and g are locally bounded in E. A direct calculation
shows that there are functions a, b : E → R+ such that
Hti,j θ(x) ≤ a(x)Hri φ(x) + b(x)Hsj g(x)
for each x ∈ Ei,j and i, j = 0, 1, . . . . Thus θ is pointwise Lipschitz in
E0,0 = E −
��
Ei,j : i, j = 0, 1, . . . and (i, j) �= (0, 0)
�
,
and for each pair (i, j) �= (0, 0), the following conditions hold:
(i) Ei,j is Hn−1+ti,j
σ-finite, and Hti,j
θ(x) < ∞ for each x ∈ Ei,j;
(ii) Hn−1+ti,j
(Ei,j) > 0 implies Hti,j
θ(x) = 0 for each x ∈ Ei,j.
This verifies that θ is an admissible map.
Lemma 2.3.6. Let A be a dyadic figure, let v ∈ Adm(A; Rn
), and define
f : A → R by the formula
f(x) :=
�
div v(x) if x ∈ int A and v is differentiable at x,
0 otherwise.
For each ε > 0 and each δ : A → R+, there is a δ-fine dyadic partition
P :=
�
(C1, x1), . . . , (Cp, xp)
�
such that [P] = A and
�
�
�
�
p
�
i=1
f(xi)|Ci| −
�
∂A
v · νA dHn−1
�
�
�
� < ε .
Proof. By Remark 2.3.3, the flux
F : B �→
�
∂B
v · νB dHn−1
is defined on the family of all figures B ⊂ A. In view of Remark 2.3.2, there
are numbers 0 ≤ sk < 1 and disjoint, possibly empty, sets Ek ⊂ A such that
v is pointwise Lipschitz in A −
�∞
i=1 Ek, and for k = 1, 2, . . . , the following
conditions hold:
(i*) Hn−1+sk
(Ek) < ∞, and Hsk
v(x) < ∞ for each x ∈ Ek;
(ii) Hn−1+sk
(Ek) > 0 implies Hsk
v(x) = 0 for each x ∈ Ek.
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-45-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 29 — #39
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2.3. Admissible maps 29
By Stepanoff’s theorem, A −
�∞
i=1 Ek is the union of disjoint sets E0 and
D ⊂ int A such that Hn
(E0) = 0 and v is differentiable at each x ∈ D. Thus
A is the union of disjoint sets D, E0, E1, . . . , and we let s0 = 1. The family
�
(Ek, sk) : k = 0, 1, . . .
�
is the disjoint union of subfamilies
�
(Ek, sk) : Hn−1+sk
(Ek) > 0
�
and
�
(Ek, sk) : Hn−1+sk
(Ek) = 0
�
,
which we enumerate as
�
(E+
i , s+
i ) : i ≥ 1
�
and
�
(E0
i , s0
i ) : i ≥ 1
�
, respectively.
For i, j ∈ N, let
E0
i,j :=
�
x ∈ E0
i : j − 1 ≤ Hs0
i
v(x) < j
�
and define t+
i := n − 1 + s+
i and t0
i := n − 1 + s0
i . Now A − D is the union
of disjoint sets E+
i and E0
i,j. Select ci > Ht+
i (E+
i ) and choose ε > 0. By
Corollary 2.1.3, there is γ : A → R+ such that for each cube C ⊂ A, the
following conditions are satisfied:
(1)
�
�f(x)|C| − F(C)
�
� ≤ ε|C| if d(C) < γ(x) for some x ∈ D ∩ C,
(2)
�
�F(C)
�
� ≤ ε2−i
c−1
i d(C)t+
i if d(C) < γ(x) for some x ∈ E+
i ∩ C,
(3)
�
�F(C)
�
� ≤ 2nj d(C)t0
i if d(C) < γ(x) for some x ∈ E0
i,j ∩ C.
Next choose δ : A → R+. With no loss of generality, we may assume that
δ ≤ γ. According to Proposition 2.2.2, there is a δ-fine dyadic partition
P :=
�
(C1, x1), . . . , (Cp, xp)
�
such that [P] = A and for κ = κ(n) > 0,
�
xk∈E+
i
d(Ck)t+
i ≤ κci and
�
xk∈E0
i,j
d(Ck)t0
i ≤ εj−1
2−i−j
.
Since f(x) = 0 for each x ∈ A − D, these inequalities, conditions (1)–(3), and
Observation 2.1.1 imply the lemma:
�
�
�
�
p
�
k=1
f(xk)|Ck| −
�
∂A
v · νA dHn−1
�
�
�
�
≤
�
xk∈D
�
�
�f(xk)|Ck| − F(Ck)
�
�
� +
�
xk∈A−D
�
�F(Ck)
�
�
≤ ε
�
xk∈D
|Ck| +
�
i≥1
� �
xk∈E+
i
�
�F(Ck)
�
� +
∞
�
j=1
�
xk∈E0
i,j
�
�F(Ck)
�
�
�
≤ ε|A| + ε
�
i≥1
�
2−i
c−1
i
�
xk∈E+
i
d(Ck)t+
i + 2n
∞
�
j=1
j
�
xk∈E0
i,j
d(Ck)t0
i
�
≤ ε|A| + εκ
∞
�
i=1
2−i
+ 2nε
∞
�
i,j=1
2−i−j
= ε
�
|A| + κ + 2n
�
.
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-46-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 30 — #40
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30 2. Divergence theorem for dyadic figures
Theorem 2.3.7. Let A be a dyadic figure. If v ∈ Adm(A; Rn
) is such that
div v belongs to L1
(A), then
�
A
div v dLn
=
�
∂A
v · νA dHn−1
.
Proof. Defining f as in Lemma 2.3.6, we have f ∈ L1
(A) and
�
A
f dLn
=
�
A
div v dLn
.
Choose ε > 0, and select a function δ : A → R+ associated with ε and f
according to Henstock’s lemma. By Lemma 2.3.6, there is a δ-fine partition
P :=
�
(C1, x1), . . . , (Cp, xp)
�
such that [P] = A and
�
�
�
�
�
A
div v dLn
−
�
∂A
v · νA dHn−1
�
�
�
� ≤
�
�
�
�
�
A
f dLn
−
p
�
i=1
f(xi)|Ci|
�
�
�
�
+
�
�
�
�
p
�
i=1
f(x)|Ci| −
�
∂A
v · νA dHn−1
�
�
�
� < 2ε.
Remark 2.3.8. Some comments are in order.
(1) The assumptions of Theorem 2.3.7 are met if v ∈ Lip(A; Rn
), since
�div v�L∞(A) ≤ nLip v.
(2) Let v(0) := 0, and v(x) := x cos |x|−n−1
for x ∈ Rn
− {0}. Then
v ∈ Adm(Rn
; Rn
), but div v does not belong to L1
(A) if A is a figure
containing 0. Still, the flux of v can be calculated from div v by an
averaging process which extends the Lebesgue integral. For a deeper
analysis of this phenomenon, we refer the interested reader to [49, 51];
also see Chapter 9 below.
(3) Assume n = 1, and let v : R → R be differentiable almost everywhere
and such that � b
a
v�
dL1
= v(b) − v(a)
for each dyadic cell [a, b] ⊂ R. Since [29, Section 2.4.3] implies
Hs
��
x ∈ R : Hsv(x) > 0
��
= 0
for each 0 ≤ s < 1, condition (ii) of Definition 2.3.1 cannot be omit-
ted. The Cantor-Vitali function (Example 9.2.5 below) and its multi-
dimensional analogue [54] provide another rationale for the definition
of admissible vector fields.
(4) It is clear that using essentially the same arguments, the divergence
theorem can be established for arbitrary figures. We employed dyadic
figures merely for convenience. Theorem 2.3.7 is only a preliminary
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-47-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 31 — #41
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2.4. Convergence of dyadic figures 31
result, sufficient for proving a satisfactory integration by parts theorem
stated below.
Theorem 2.3.9 (Integration by parts). Let Ω ⊂ Rn
be an open set, and let
both v : Ω → Rn
and g : Ω → R be locally bounded and pointwise Lipschitz
almost everywhere in Ω. Assume div v ∈ L1
loc(U) and Dg ∈ L1
loc(Ω; Rn
). If
gv ∈ Adm(Ω; Rn
) and spt (gv) � Ω, then
�
Ω
g(x) div v(x) dx = −
�
Ω
Dg(x) · v(x) dx.
Proof. Let w = gv, and let A ⊂ Ω be a dyadic figure with spt w ⊂ int A.
By the assumptions, v and g are measurable and bounded in A. Thus
div w = g div v + Dg · v
belongs to L1
(A). Since w � ∂A = 0, Theorem 2.3.7 yields
0 =
�
A
div w(x) dx =
�
A
g(x) div v(x) dx +
�
A
Dg(x) · v(x) dx. (∗)
Let x ∈ Ω − A. Then div w(x) = 0, since w vanishes in the open setΩ − A.
Thus g(x) div v(x) = −Dg(x) · v(x) and either g(x) = 0 or v(x) = 0. This
shows that spt (g div v) and spt (Dg · v) are subsets of A, and the theorem
follows from (∗).
Remark 2.3.10. The integration by parts theorem is usually applied when
both g and v are admissible, and either g or v has compact support contained
in Ω; see Proposition 2.3.5.
2.4. Convergence of dyadic figures
As figures are too specialized for applications, it is desirable to extend the divergence
theorem to a larger family of sets. With the sole purpose of enhancing intuition, we describe
the first step of the most obvious approach to this problem. Our main results will be
obtained from less obvious but more efficient ideas of R. Caccioppoli [14] and E. De Giorgi
[18, 19].
Lemma 2.4.1. Let {Ai} be a sequence of measurable set, and let E ⊂ Rn be any set. If
lim |E � Ai| = 0, then E is measurable and for each f ∈ L1(Rn),
lim
�
Ai
f(x) dx =
�
E
f(x) dx.
Proof. Since lim |E − Ai| = lim |Ai − E| = 0, passing to a subsequence if necessary, we
may assume that |E − Ai| ≤ 2−i and |Ai − E| ≤ 2−i for i = 1, 2, . . . . Letting
I := lim inf Ai =
∞
�
j=1
∞
�
i=j
Aj and S := lim sup Ai =
∞
�
j=1
∞
�
i=j
Aj,
we infer |E − I| = |S − E| = 0. As I ⊂ S implies |E � I| = |E � S| = 0, the set
E is measurable and lim �χE − χAi
�L1(Rn) = 0. Any subsequence {Bi} of {Ai} has a
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-48-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 32 — #42
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32 2. Divergence theorem for dyadic figures
subsequence {Ci} such that lim χCi
= χE almost everywhere. Thus for f ∈ L1(Rn), the
dominated convergence theorem yields
lim
�
�
�
�
�
E
f(x) dx −
�
Ci
f(x) dx
�
�
�
� ≤ lim
�
Rn
�
�χE(x) − χCi
(x)
�
� ·
�
�f(x)
�
� dx = 0.
The lemma follows from the arbitrariness of {Bi}.
Recall that the family of all dyadic figures in Rn is denoted by DF. We say that a
sequence {Ai} in DF converges to a set E ⊂ Rn if the following conditions are satisfied:
(i) Each Ai is contained in a fixed compact set K ⊂ Rn.
(ii) lim |Ai � E| = 0 and sup Hn−1(∂Ai) < ∞.
By Lemma 2.4.1, the set E is measurable.
Given dyadic figures A and B, we define nonoverlapping dyadic figures
A � B := cl (A − B) and A ⊙ B = cl
�
int (A ∩ B)],
and observe that A = (A � B) ∪ (A ⊙ B).
Proposition 2.4.2. Let F be the flux of v ∈ C(Rn; Rn), and let {Ai} be a sequence in
DF converging to a set E ⊂ Rn. There exists a finite limit
�
F(E) := lim F(Ai),
which does not depend on the choice of the sequence {Ai}.
Proof. Let K ⊂ Rn be a compact set containing all figures Ai, and let c = sup Hn−1(∂Ai).
Choose ε > 0, and use the Stone-Weierstrass theorem [62, Theorem 7.32] to find a vector
field w ∈ C1(Rn; Rn) such that �v − w�L∞(K;Rn) ≤ ε. According to Theorem 2.3.7,
�
�F(Ai � Aj)
�
� ≤
�
�
�
�
�
∂(Ai�Aj )
(v − w) · νAi�Aj
dHn−1
�
�
�
�
+
�
�
�
�
�
∂(Ai�Aj )
w · νAi�Aj
dHn−1
�
�
�
�
≤ εHn−1
�
∂(Ai � Aj)
�
+
�
�
�
�
�
Ai�Aj
div w(x) dx
�
�
�
�
≤ 2cε + �div w�L∞(K)|Ai � Aj|;
since ∂(Ai � Aj) ⊂ ∂Ai ∪ ∂Aj. By Observation 2.1.1,
�
�F(Ai) − F(Aj)
�
� =
�
�
�
�
F(Ai � Aj) + F(Ai ⊙ Aj)
�
−
�
F(Aj � Ai) + F(Aj ⊙ Ai)
��
�
�
≤
�
�F(Ai � Aj)
�
� +
�
�F(Aj � Ai)
�
�
≤ �div w�L∞(K)
�
|Ai � Aj| + |Aj � Ai|
�
+ 4cε
= �div w�L∞(K)|Ai � Aj| + 4cε.
It follows that
�
F(Ai)
�
is a Cauchy sequence. The value
�
F(E) := lim F(Ai)
does not depend on the sequence {Ai} of dyadic figures converging to E. Indeed, if {Bi}
is another sequence of dyadic figures converging to E, then so does the interlaced sequence
{A1, B1, A2, B2, . . . }, and consequently
lim F(Ai) = lim F(Bi).
Denote by DF the family of all sets E ⊂ Rn, necessarily measurable, for which there
is a sequence {Ai} in DF converging to E. Clearly DF ⊂ DF, and since
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-49-320.jpg)
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“Book˙2011” — 2012/2/26 — 9:58 — page 36 — #46
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36 3. Removable singularities
for each sequence {ϕi} in D(Ω; C) that converges to zero in the sense of test
functions. The complex linear space of all distributions is denoted by D�
(Ω; C).
The real linear space D�
(Ω) is defined analogously.
Remark 3.1.1. In Example 3.6.5 below we define a locally convex topology S in the spaces
D(Ω) of real-valued test functions so that the space D�(Ω) of distributions is the dual space
of
�
D(Ω), S). The reader familiar with complex locally convex spaces will recognize instantly
that a similar topology can be defined in the space D(Ω; C) of complex-valued test functions
[64, Section 6.2].
Example 3.1.2. Let f ∈ L1
loc(Ω; C), let µ be a Radon measure in Ω, and let
v ∈ L1
loc(Ω; Rn
). The linear functionals
Lf : ϕ �→
�
Ω
fϕ dx : D(Ω; C) → C, (1)
Lµ : ϕ �→
�
Ω
ϕ dµ : D(Ω; C) → C, (2)
Fv : ϕ �→ −
�
Ω
v · Dϕ dx : D(Ω) → R (3)
are examples of distributions. Distribution Fv is called the distributional
divergence of v, since if v ∈ C1
(Ω; Rn
) integration by parts yields
Fv(ϕ) =
�
Ω
ϕ div v dx = Ldiv v(ϕ)
for each test function ϕ ∈ D(Ω).
Let α be a multi-index. If f ∈ C|α|
(Ω; C), then repeated applications of
the integration by parts theorem yield
�LDαf , ϕ� =
�
Ω
ϕ(x)Dα
f(x) dx
= (−1)|α|
�
Ω
f(x)Dα
ϕ(x) dx = (−1)|α|
�Lf , Dα
ϕ�
for each ϕ ∈ D(Ω; C). Since for any distribution L, the linear functional
ϕ �→ (−1)|α|
�L, Dα
ϕ� : D(Ω; C) → C
is a distribution, the previous identity suggests to define a distribution Dα
L
by the formula
�Dα
L,ϕ � := (−1)|α|
�L, Dα
ϕ�
for each test function ϕ ∈ D(Ω; C). Observe that
Dα
Lf = LDαf
whenever f ∈ C|α|
(Ω; C). Additional information about test functions and
distributions can be found in many standard textbooks, for instance in [64,
Chapter 6] or [27, Chapter 5].
© 2012 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-53-320.jpg)
![reason, adopt this because their fathers before them did so. It is
urged by intelligent naturalists that it is utterly useless, as bees have
no sense of hearing. What does the clamour mean,—whence
derived?
B. B.
Pembroke.
Sally Lunn (Vol. v., p. 371.).
—In reply to the Query, "Is anything known of Sally Lunn? is she a
personage or a myth?" I refer your inquirer to Hone's Every-day
Book, vol. ii. p. 1561.:
"The bun so fashionable, called the Sally Lunn, originated
with a young woman of that name at Bath, about thirty years
ago." [This was written in 1826.] "She first cried them in a
basket, with a white cloth over it, morning and evening.
Dalmer, a respectable baker and musician, noticed her,
bought her business, and made a song and set it to music in
behalf of Sally Lunn. This composition became the street
favourite, barrows were made to distribute the nice cakes,
Dalmer profited thereby and retired, and to this day the Sally
Lunn Cake claims pre-eminence in all the cities of England."
J. R. W.
Bristol.
Baxter's Pulpit (Vol. v., p. 363.).
—An engraving of Baxter's pulpit will be found in a work entitled
Footsteps of our Forefathers: what they suffered and what they
sought. By James G. Miall, 1851, p. 232.
J. R. W.
Bristol.
Lothian's Scottish Historical Maps (Vol. v., p. 371.).](https://image.slidesharecdn.com/2035708-250527101042-121307db/85/The-Divergence-Theorem-And-Sets-Of-Finite-Perimeter-Washek-F-Pfeffer-61-320.jpg)
The Divergence Theorem And Sets Of Finite Perimeter Washek F Pfeffer
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- 5. The Divergence Theorem and Sets of Finite Perimeter K14657_FM.indd 1 3/14/12 5:31 PM © 2012 by Taylor & Francis Group, LLC
- 6. PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS EDITORIAL BOARD Earl J. Taft Rutgers University Piscataway, New Jersey Zuhair Nashed University of Central Florida Orlando, Florida Jane Cronin Rutgers University S. Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University Anil Nerode Cornell University Freddy van Oystaeyen University of Antwerp, Belgium Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universität Siegen K14657_FM.indd 2 3/14/12 5:31 PM © 2012 by Taylor & Francis Group, LLC
- 7. MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS Recent Titles Marwan Moubachir and Jean-Paul Zolésio, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions (2006) Alfred Geroldinger and Franz Halter-Koch, Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory (2006) Kevin J. Hastings, Introduction to the Mathematics of Operations Research with Mathematica®, Second Edition (2006) Robert Carlson, A Concrete Introduction to Real Analysis (2006) John Dauns and Yiqiang Zhou, Classes of Modules (2006) N. K. Govil, H. N. Mhaskar, Ram N. Mohapatra, Zuhair Nashed, and J. Szabados, Frontiers in Interpolation and Approximation (2006) Luca Lorenzi and Marcello Bertoldi, Analytical Methods for Markov Semigroups (2006) M. A. Al-Gwaiz and S. A. Elsanousi, Elements of Real Analysis (2006) Theodore G. Faticoni, Direct Sum Decompositions of Torsion-Free Finite Rank Groups (2007) R. Sivaramakrishnan, Certain Number-Theoretic Episodes in Algebra (2006) Aderemi Kuku, Representation Theory and Higher Algebraic K-Theory (2006) Robert Piziak and P. L. Odell, Matrix Theory: From Generalized Inverses to Jordan Form (2007) Norman L. Johnson, Vikram Jha, and Mauro Biliotti, Handbook of Finite Translation Planes (2007) Lieven Le Bruyn, Noncommutative Geometry and Cayley-smooth Orders (2008) Fritz Schwarz, Algorithmic Lie Theory for Solving Ordinary Differential Equations (2008) Jane Cronin, Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition (2008) Su Gao, Invariant Descriptive Set Theory (2009) Christopher Apelian and Steve Surace, Real and Complex Analysis (2010) Norman L. Johnson, Combinatorics of Spreads and Parallelisms (2010) Lawrence Narici and Edward Beckenstein, Topological Vector Spaces, Second Edition (2010) Moshe Sniedovich, Dynamic Programming: Foundations and Principles, Second Edition (2010) Drumi D. Bainov and Snezhana G. Hristova, Differential Equations with Maxima (2011) Willi Freeden, Metaharmonic Lattice Point Theory (2011) Murray R. Bremner, Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications (2011) Clifford Bergman, Universal Algebra: Fundamentals and Selected Topics (2011) A. A. Martynyuk and Yu. A. Martynyuk-Chernienko, Uncertain Dynamical Systems: Stability and Motion Control (2012) Washek F. Pfeffer, The Divergence Theorem and Sets of Finite Perimeter (2012) K14657_FM.indd 3 3/14/12 5:31 PM © 2012 by Taylor & Francis Group, LLC
- 8. Washek F. Pfeffer University of California, Davis University of Arizona, Tucson USA The Divergence Theorem and Sets of Finite Perimeter K14657_FM.indd 5 3/14/12 5:31 PM © 2012 by Taylor & Francis Group, LLC
- 9. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20120312 International Standard Book Number-13: 978-1-4665-0721-0 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a pho- tocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2012 by Taylor & Francis Group, LLC
- 10. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page vii — #1 ✐ ✐ ✐ ✐ ✐ ✐ To Lida for her love and a lifetime of companionship © 2012 by Taylor & Francis Group, LLC
- 11. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page ix — #3 ✐ ✐ ✐ ✐ ✐ ✐ Contents Preface xiii Part 1. Dyadic figures 1 1. Preliminaries 3 1.1. The setting 3 1.2. Topology 7 1.3. Measures 9 1.4. Hausdorff measures 13 1.5. Differentiable and Lipschitz maps 16 2. Divergence theorem for dyadic figures 21 2.1. Differentiable vector fields 21 2.2. Dyadic partitions 24 2.3. Admissible maps 27 2.4. Convergence of dyadic figures 31 3. Removable singularities 35 3.1. Distributions 35 3.2. Differential equations 37 3.3. Holomorphic functions 38 3.4. Harmonic functions 39 3.5. The minimal surface equation 39 3.6. Injective limits 41 Part 2. Sets of finite perimeter 47 4. Perimeter 49 4.1. Measure-theoretic concepts 49 4.2. Essential boundary 51 4.3. Vitali’s covering theorem 53 4.4. Density 54 4.5. Definition of perimeter 56 ix © 2012 by Taylor & Francis Group, LLC
- 12. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page x — #4 ✐ ✐ ✐ ✐ ✐ ✐ x Contents 4.6. Line sections 58 4.7. Lipeomorphisms 68 5. BV functions 73 5.1. Variation 73 5.2. Mollification 76 5.3. Vector valued measures 79 5.4. Weak convergence 86 5.5. Properties of BV functions 92 5.6. Approximation theorem 98 5.7. Coarea theorem 101 5.8. Bounded convex domains 106 5.9. Inequalities 110 5.10. Lipschitz maps 117 6. Locally BV sets 121 6.1. Dimension one 121 6.2. Besicovitch’s covering theorem 123 6.3. The reduced boundary 126 6.4. Blow-up 131 6.5. Perimeter and variation 137 6.6. Properties of BV sets 142 6.7. Approximating by figures 146 Part 3. The divergence theorem 149 7. Bounded vector fields 151 7.1. Approximating from inside 151 7.2. Relative derivatives 155 7.3. The critical interior 158 7.4. The divergence theorem 160 7.5. Lipschitz domains 166 7.6. BV vector fields 179 8. Unbounded vector fields 181 8.1. Minkowski contents 181 8.2. Controlled vector fields 185 8.3. Integration by parts 190 9. Mean divergence 193 9.1. The derivative 193 9.2. The critical variation 197 © 2012 by Taylor & Francis Group, LLC
- 13. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page xi — #5 ✐ ✐ ✐ ✐ ✐ ✐ Contents xi 10. Charges 205 10.1. Continuous vector fields 205 10.2. Localized topology 207 10.3. Locally convex spaces 209 10.4. Duality 212 10.5. The space BVc(Ω) 213 10.6. Streams 216 11. The divergence equation 219 11.1. Background 219 11.2. Solutions in Lp (Ω; Rn ) 221 11.3. Continuous solutions 224 Bibliography 231 List of symbols 235 Index 237 © 2012 by Taylor & Francis Group, LLC
- 14. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 243 — #253 ✐ ✐ ✐ ✐ ✐ ✐ Washek F. Pfeffer is Professor Emeritus of Mathematics at the University of California in Davis. He was born in Prague, Czech Republic, where he studied mathematics at Charles University (1955–60). He immigrated to the United States in 1965, and in 1966 received his Ph.D. from the University of Mary- land in College Park. Dr. Pfeffer has worked at the Czechoslovak Academy of Sciences in Prague, and has taught at the Royal Institute of Technology in Stockholm, George Washington University, University of California in Berke- ley, University of Ghana in Accra, and King Fahd University in Dhahran, Saudi Arabia. In 1994–95 he was a Fulbright Lecturer at Charles University. His primary research areas are analysis and topology. Dr. Pfeffer is a member of the American and Swedish Mathematical Societies, and an honorary mem- ber of the Academic Board of the Center for Theoretical Study at Charles University. Presently, he is a Research Associate in the Mathematics Depart- ment of the University of Arizona. He has written the books Integrals and Measures (Marcel Dekker, 1977), The Riemann Approach to Integration, and Derivation and Integration (Cambridge University Press, 1993 and 2001). © 2012 by Taylor & Francis Group, LLC
- 15. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page xiii — #7 ✐ ✐ ✐ ✐ ✐ ✐ Preface The divergence theorem and the resulting integration by parts formula belong to the most frequently used tools of mathematical analysis. In its elementary form, that is for smooth vector fields defined in a neighborhood of some simple geometric object such as rectangle, cylinder, ball, etc., the divergence theorem is presented in many calculus books. Its proof is obtained by a simple appli- cation of the one-dimensional fundamental theorem of calculus and iterated Riemann integration. Appreciable difficulties arise when we consider a more general situation. Employing the Lebesgue integral is essential, but it is only the first step in a long struggle. We divide the problem into three parts. (1) Extending the family of vector fields for which the divergence theorem holds on simple sets. (2) Extending the family of sets for which the divergence theorem holds for Lipschitz vector fields. (3) Proving the divergence theorem when the vector fields and sets are extended simultaneously. Of these problems, part (2) is unquestionably the most complicated. While many mathematicians contributed to it, the Italian school represented by Caccioppoli, De Giorgi, and others obtained a complete solution by defining the sets of bounded variation (BV sets). A major contribution to part (3) is due to Federer, who proved the divergence theorem for BV sets and Lipschitz vector fields. While parts (1)–(3) can be combined, treating them separately illuminates the exposition. We begin with sets that are locally simple — finite unions of dyadic cubes, called dyadic figures. Combining ideas of Henstock and McShane with a combinatorial argument of Jurkat, we establish the divergence theorem for very general vector fields defined on dyadic figures. The proof involves only basic properties of the Lebesgue integral and Hausdorff measures. An easy corollary of the divergence theorem is a powerful integration by parts formula. It yields results on removable sets for the Cauchy-Riemann, Laplace, and min- imal surface equations. The next goal is to move from dyadic figures to BV sets. To enhance the intuition, our starting point is the geometric definition of perimeter. The perimeter of a set is the codimension one Hausdorff measure of its essential xiii © 2012 by Taylor & Francis Group, LLC
- 16. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page xiv — #8 ✐ ✐ ✐ ✐ ✐ ✐ xiv Preface boundary. Several properties of sets with finite perimeter are derived directly from the definition. Deeper results rest on the equivalent analytic definition. Following the standard presentation, we say that an integrable function has bounded variation, or is a BV function, if its distributional gradient is a vector-valued measure of finite variation. The variation of a BV function is defined as the variation of its distributional gradient. A set whose indicator is a BV function is called a BV set. The variation of a BV set is the variation of its indicator. Although we are mainly interested in BV sets, it is neither possible nor desirable to separate them from BV functions. It is often easier to prove a result about BV functions first, and state the corresponding result about BV sets as a corollary. The link between BV sets and BV functions is the coarea formula, which connects the variation of a function with that of its level sets. Our objective is to show the equivalence of the geometric and analytic definitions by equating the perimeter of a measurable set with its variation. A variety of useful results concerning BV sets follows from the interplay between the two definitions. Throughout, we derive properties of BV functions directly from the definition, without referring to corresponding properties of Sobolev spaces. Sobolev spaces are not discussed in this text, and no a priori knowledge about them is required. Once the BV sets are defined and their main properties established, it is relatively easy to apply the divergence theorem we proved for dyadic figures to BV sets. The main tool, due to Giacomelli and Tamanini, consists of approximating arbitrary BV sets by their BV subsets with special properties. At the end, we extend the divergence theorem to a family of unbounded vector fields with controlled growth. We pay particular attention to continuous vector fields and their weak divergence. Elaborating on ideas of Bourgain and Brezis, we characterize the distributions F for which the divergence equation div v = F has a continuous weak solution — a recent joint work of T. De Pauw and the author. All of our results and proofs rely entirely on the Lebesgue integral. No exotic integrals, akin to the generalized Riemann integral of Henstock and Kurzweil, are involved. Notwithstanding, some techniques we use are inspired by investigations of these integrals. We strove to give complete and detailed proofs of all our claims. Only a few standard facts are quoted without proofs, in which case we always provide precise references. The book has three parts, roughly corresponding to parts (1)–(3) listed above. We trust that the titles of the chapters and sections are sufficiently descriptive. Results and comments we consider marginal are presented in small print. However, marginal does not mean unimportant; a useful enhancement of the main text can be found in the small print. © 2012 by Taylor & Francis Group, LLC
- 17. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page xv — #9 ✐ ✐ ✐ ✐ ✐ ✐ Preface xv The first two chapters, which deal with dyadic figures, are quite elemen- tary. Except for very basic properties of Hausdorff measures, they should be accessible to the beginning graduate students. The rest of the book presup- poses the knowledge equivalent to the first year graduate course in analysis. In addition, some familiarity with Hausdorff measures and distributions is ex- pected. Rudimentary results from functional analysis are employed in the last two chapters. Our presentation owes much to the excellent textbooks [29, 75] and monographs [1, 33], which can serve as useful references. During the preparation of this text I largely benefited from discussions with L. Ambrosio, P. Bouafia, G.D. Chakerian, T. De Pauw, D.B. Fuchs, N. Fusco, R.J. Gardner, G. Gruenhage, Z. Nashed, M. Šilhavý, S. Solecki, V. Sverak, B.S. Thomson, and M. Torres. I am obliged to W.G. McCallum who offered me a position of Research Associate in the Mathematics Depart- ment of the University of Arizona; it gave me access to university facilities, in particular to the university library. Editorial help provided by the publisher was invaluable. In this regard my thanks belong to K. Craig, M. Dimont, S. Kumar, S. Morkert, and R. Stern. W.F.P. Tucson, Arizona February 2012 © 2012 by Taylor & Francis Group, LLC
- 18. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 1 — #11 ✐ ✐ ✐ ✐ ✐ ✐ Part 1 Dyadic figures © 2012 by Taylor & Francis Group, LLC
- 20. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 3 — #13 ✐ ✐ ✐ ✐ ✐ ✐ Chapter 1 Preliminaries We establish the notation and terminology, and present some basic facts that will be used throughout the book. Several well-known theorems are stated without proofs; however, those results for which we found no convenient ref- erences are proved in detail. In general, the reader is expected to have some prior knowledge of the concepts introduced in this chapter. 1.1. The setting The sets of all integers and of all positive integers are denoted by Z and N, respectively. Symbols Q, R, and C denote, respectively, the sets of all rational, real, and complex numbers. The sets of all positive real numbers and of all positive rationals numbers are denoted by R+ and Q+, respectively. Unless specified otherwise, by a number we always mean a real number. Elements of R := R ∪ {±∞} are called the extended real numbers. In R we consider the usual order and topology, and define the following algebraic operations: a + ∞ : = +∞ + a := +∞ for a > −∞, a − ∞ : = −∞ + a := −∞ for a < +∞, a · (±∞) : = ±∞ if a > 0, ∓∞ if a < 0, 0 if a = 0. At various places we write P := Q instead of P = Q to stress the fact that P is defined as equal to Q. Throughout, the symbol ∞ stands for +∞. Unless specified otherwise, ε → 0 means ε → 0+. Finite and countably infinite sets are called countable. We say that a family E of sets covers a set E, or is a cover of E, if E ⊂ � E. For any pair of sets A and B, the set A � B := (A − B) ∪ (B − A) = A ∪ B − A ∩ B is called the symmetric difference of A and B. 3 © 2012 by Taylor & Francis Group, LLC
- 21. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 4 — #14 ✐ ✐ ✐ ✐ ✐ ✐ 4 1. Preliminaries By a function we always mean an extended real-valued function. A finite function is real-valued. If a function f equals identically to c ∈ R, we write f ≡ c. When no confusion can arise, the same symbol denotes a function f defined on a set A and its restriction f � B to a set B ⊂ A. For a function f defined on a set E and t ∈ R, we let {f > t} := � x ∈ E : f(x) > t � , and define the sets {f ≥ t}, {f < t}, etc. similarly. The set {f = 0} is called the null set of f. Further, we let f+ := max{f, 0} and f− := max{−f, 0}, and note that f = f+ − f− and |f| = f+ + f− . The value of f at x ∈ E is denoted interchangeably by f(x), f[x], and �f, x�. For m ∈ N, and x := (ξ1, . . . , ξm) and y := (η1, . . . , ηm) in Rm , x · y := m � i=1 ξiηi and |x| := √ x · x. In Rm we use exclusively the Euclidean metric induced by the norm |x|. The diameter, closure, interior, and boundary of a set E ⊂ Rm are denoted by d(E), cl E, int E, and ∂E, respectively. The distance between sets A, B ⊂ Rm is denoted by dist(A, B), or dist(x, B) if A = {x} is a singleton. Given E ⊂ Rm and r ∈ R+, we let U(E, r) : = � x ∈ Rm : dist(x, E) < r � , B(E, r) : = � x ∈ Rm : dist(x, E) ≤ r � . If E = {x} is a singleton, the sets U(x, r) := U � {x}, r � and B(x, r) := B � {x}, r � are, respectively, the open and closed ball in Rm of radius r centered at x. For a pair of sets A, B ⊂ Rm , the symbol A � B indicates that cl A is a compact subset of int B. Given E ⊂ Rm and s ∈ N, we denote by C(E; Rs ) the linear space of all continuous maps φ : E → Rs . We let C(E) := C(E; R), and note that according to this definition, all continuous function are real-valued. The Urysohn function associated with a pair A, B of closed disjoint sub- sets of Rm is a function uA,B ∈ C(Rm ) defined by the formula uA,B(x) := dist (x, B) dist (x, A) + dist (x, B) , x ∈ Rm . (1.1.1) Theorem 1.1.1 (Tietze). Let C ⊂ Rm be a closed set. Each continuous map φ : C → Rs has a continuous extension ψ : Rm → Rs . © 2012 by Taylor & Francis Group, LLC
- 22. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 5 — #15 ✐ ✐ ✐ ✐ ✐ ✐ 1.1. The setting 5 Proof. As φ has a continuous extension if and only if each of its coordinates does, it suffices to show that every continuous function f : C → R has a continuous extension g : Rm → R. Now R is homeomorphic to the open interval (−1, 1), e.g., via the strictly increasing continuous function ϕ : x �→ 2 π tan−1 x : R → (−1, 1). Hence we may assume that f : C → (−1, 1). Let uC1−,C1+ be the Urysohn function associated with C1± := � x ∈ C : ±f(x) ≥ 1/3 � . If g1 := 1/3 − (2/3)uC1−,C1+ , then � �g1(x) � � ≤ 3−1 for all x ∈ Rm , � �f(x) − g1(x) � � ≤ 3−1 for all x ∈ C. Next let f1 = f −g1, and let uC2−,C2+ be the Urysohn function associated with C2± := � x ∈ C : ±f1(x) ≥ 1/32 � . If g2 := 1/32 − (2/32 )uC2−,C2+ , then � �g2(x) � � ≤ 3−2 for all x ∈ Rm , � �f(x) − g1(x) − g2(x) � � ≤ 3−2 for all x ∈ C. Proceeding by recursion, we define functions gk ∈ C(Rm ) such that |gk| ≤ 3k and � �f − �k j=1(gj � C) � � ≤ 3k for k = 1, 2, . . . . It is clear that g := �∞ k=1 gk belongs to C(Rm ) and extends f. Corollary 1.1.2. Let Ω ⊂ Rm be an open set, and let C ⊂ Ω be a closed set. Each continuous map φ : C → Rs has a continuous extension θ : Rm → Rs such that cl {θ �= 0} ⊂ Ω. Proof. By Titze’s theorem φ has a continuous extension ψ : C → Rs . Let f = uC,Rm−Ω be the Urysohn function associated with C and Rm − Ω, and let D = {f ≤ 1/2}. Note C ⊂ Rm − D ⊂ cl (Rm − D) ⊂ Ω. If uC,D is the Urysohn function associated with C and D, then θ = uC,Dψ is the desired extension. IfΩ ⊂ Rm is an open set and k ∈ N, then Ck (Ω; Rs ) denotes the linear space of all maps φ = (f1, . . . , fs) from Ω to Rs such that each fi : Ω → R has continuous partial derivatives of orders less than or equal to k. We let C∞ (Ω; Rs ) = ∞ � k=1 Ck (Ω; Rs ), and refer to elements of Ck (Ω; Rs ) and C∞ (Ω; Rs ), respectively, as Ck and C∞ maps from Ω to Rs . Instead of C(E; R), Ck (Ω; R), and C∞ (Ω; R), we write C(E), Ck (Ω), and C∞ (Ω), respectively. The elements of Ck (Ω) and C∞ (Ω) are called, respectively, the Ck and C∞ functions defined on Ω. © 2012 by Taylor & Francis Group, LLC
- 23. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 6 — #16 ✐ ✐ ✐ ✐ ✐ ✐ 6 1. Preliminaries Let E ⊂ Rm and φ : E → Rs . The indicator χE of E and the zero extension φ of φ are defined by the formulae χE(x) := � 1 if x ∈ E, 0 if x ∈ Rm − E, φ(x) := � φ(x) if x ∈ E, 0 if x ∈ Rm − E, respectively. The support of φ is the set spt φ := cl {φ �= 0}. LetΩ ⊂ Rm be an open set. The linear space of all φ ∈ C(Ω; Rs ) with spt φ � Ω is denoted by Cc(Ω; Rs ); the spaces Cc(Ω), Ck c (Ω; Rs ), C∞ c (Ω; Rs ), etc., are defined similarly. We always identify φ ∈ Cc(Ω; Rs ) with its zero extension φ ∈ Cc(Rm ; Rs ). This simple convention, which will cause no con- fusion, legitimizes the inclusions Cc(U; Rs ) ⊂ Cc(Ω; Rs ) ⊂ Cc(Rm ; Rs ), Ck c (U; Rs ) ⊂ Ck c (Ω; Rs ) ⊂ Ck c (Rm ; Rs ) (1.1.2) where U ⊂ Ω is an open set and k = 1, 2, . . . , ∞. Throughout this book, the ambient space is Rn where n ≥ 1 is a fixed integer. By {e1, . . . , en} we denote the standard base in Rn , i.e., ei := (0, . . . , 1 i-th place , . . . , 0), i = 1, . . . , n. The projection in the direction of ei is the linear map πi : x �→ � j�=i (x · ej)ej : Rn → Rn . As the setΠ i := πi(Rn ) is a linear subspace of Rn with bases e1, . . . , ei−1, ei+1, . . . , en, it is isometric to Rn−1 . For each x ∈ Πi, the set π−1 i (x) is isometric to R. Thus whenever convenient, we tacitly identify the spaceΠ i with Rn−1 , and the set π−1 i (x) with R. Note that if n = 1, then π1(x) = 0 for each x ∈ R1 , and hence π1(R1 ) = {0}. A cell in Rn is the set A := �n j=1[aj, bj] where aj < bj are real numbers. If b1 − a1 = · · · = bn − an, the cell A is called a cube. A figure is a finite, possibly empty, union of cells. A k-cube is a cube n � j=1 � ij2−k , (ij + 1)2−k � where k and i1, . . . , in are integers. The family of all k-cubes is denoted by Dk, and the elements of the union DC := � k∈Z Dk are called dyadic cubes. A dyadic figure is a finite, possibly empty, union of dyadic cubes. The family of all dyadic figures in Rn is denoted by DF. At places we employ unspecified positive constants depending on certain parameters, such as the dimension n. If κ is a constant depending only on © 2012 by Taylor & Francis Group, LLC
- 24. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 7 — #17 ✐ ✐ ✐ ✐ ✐ ✐ 1.2. Topology 7 parameters p1, . . . , pk, we write κ = κ(p1, . . . , pk). With a few exceptions, we use no universal constants. Symbols denoting constants are tied to the context: distinct constants appearing in different contexts are often denoted by the same symbol. 1.2. Topology All topologies considered in this book are assumed to be Hausdorff. If T and S are topologies in a set X and T ⊂ S, we say that S is larger than T, or equivalently that T is smaller than S. In a topological space (X, T), the closure of E ⊂ X is denoted by cl TE. Unless specified otherwise, each Y ⊂ X is given the subspace topology. Let X be a topological space. A set E ⊂ X is a Gδ set if it is the intersection of countably many open subsets of X. Borel sets in X are elements of the smallest σ-algebra in X containing all open subsets of X. A map φ from X to a topological space Y is called Borel measurable, or merely Borel, if φ−1 (B) := � x ∈ X : φ(x) ∈ B � is a Borel subset of X for every Borel set B ⊂ Y . A subset E of a topological space X is called sequentially closed if each sequence {xk} in E that converges in X converges to x ∈ E. Each closed subset of X is sequentially closed but not vice versa; see Example 1.2.1 below. If the converse is true, i.e., if every sequentially closed set E ⊂ X is closed, the space X is called sequential. All closed and all open subsets of a sequential space are sequential. A map φ from a sequential space X to any topological space Y is continuous whenever lim φ(xk) = φ(lim xk) for every convergent sequence {xk} in X. Each first countable space is se- quential, but the converse is false; see Example 10.3.4. Example 1.2.1. Let ω1 be the first uncountable ordinal, and let X be the space of all ordinals smaller than or equal to ω1 equipped with the order topology. The set E = X − {ω1} is sequentially closed but not closed. Unless specified otherwise, a linear space is a linear space over R. Let X be a linear space. The zero element of X is denoted by 0, and A + B := {x + y : x ∈ A and y ∈ B} and tA := {tx : x ∈ A} for A, B ⊂ X and t ∈ R. As usual −A := (−1)A and x + A := {x} + A for x ∈ X. A set C ⊂ X is called © 2012 by Taylor & Francis Group, LLC
- 25. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 8 — #18 ✐ ✐ ✐ ✐ ✐ ✐ 8 1. Preliminaries • absorbing if X = � {tC : t ∈ R}, • symmetric if −C = C, • convex if tC + (1 − t)C ⊂ C for each 0 ≤ t ≤ 1. The linear hull, or convex hull, of a set E ⊂ X is, respectively, the intersection of all linear subspaces of X containing E, or the intersection of all convex subsets of X containing E. A topology in X for which the maps (x, y) �→ x + y : X × X → X and (t, x) �→ tx : R × X → X are continuous is called linear. Since each linear topology is induced by a uniformity [28, Example 8.1.17], all topological linear spaces are completely regular [28, Theorem 8.1.21]. A locally convex topology is a linear topology that has a neighborhood base at zero consisting of convex sets. A linear space equipped with a linear, or locally convex, topology is called a topological linear space, or a locally convex space, respectively. In this book we encounter only locally convex spaces. A seminorm in X is a functional p : X → R such that p(x + y) ≤ p(x) + p(y) and p(tx) = |t|p(x) for all x, y ∈ X and each t ∈ R. Observe that p(0) = 0 ≤ p(x) for each x ∈ X. A norm in X is a seminorm p such that p(x) = 0 implies x = 0. A family P of seminorms is called separating if p(x) = 0 for all p ∈ P implies x = 0. A separating family P defines a locally convex topology in X; the neighborhood base at zero is given by convex symmetric sets Up1,...,pk;ε := � x ∈ X : max � p1(x), . . . , pk(x) � < ε � where p1, . . . , pk are in P and ε > 0. Conversely, each locally convex topology in X is induced by a separating family P of seminorms [64, Remark 1.38, (b)]. The separating property of P guarantees that the topology defined by P is Hausdorff. A locally convex topology induced by a countable separating family {pk : k ∈ N} of seminorms is metrizable; for instance, by the metric ρ(x, y) := ∞ � k=1 2−k pk(x − y) 1 + pk(x − y) . A Fréchet space is a completely metrizable locally convex space. Even if a topology in X is defined by an uncountable family of seminorms, there may exist another family of seminorms in X that is countable and defines the same topology. The next example illustrates the situation. Example 1.2.2. LetΩ ⊂ Rm be an open set. The topology T of locally uniform convergence in C(Ω; Rs ) is defined by the seminorms pK(φ) := sup x∈K � �φ(x) � � © 2012 by Taylor & Francis Group, LLC
- 26. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 9 — #19 ✐ ✐ ✐ ✐ ✐ ✐ 1.3. Measures 9 where K ⊂ Ω is a compact set and φ ∈ C(Ω; Rs ). Each x ∈ Ω ∩ Qm has an open neighborhood Ux � Ω. Organize {Ux : x ∈ Ω ∩ Qm } into a sequence U1, U2, . . . , and let Vj = �j i=1 Ui. Since Vj � Ω, and since each compact set K ⊂ Ω is contained in some Vj, it is clear that T is defined by the seminorms qj(φ) := sup x∈Vj � �φ(x) � �, j = 1, 2 . . . . It follows that T is the metrizable topology of uniform convergence on the sets Vj. Thus T is complete, and � C(Ω; Rm ), T � is a Fréchet space. Let (X, T) be a locally convex space. A set E ⊂ X is called bounded if for each convex neighborhood U of zero there is t > 0 such that E ⊂ tU. Every compact set E ⊂ X is bounded. In general, E ⊂ X is bounded if and only if lim tkxk = 0 whenever {xk} is a sequence in E and {tk} is a sequence in R converging to zero [64, Theorem 1.30]. If the topology T is induced by a family P of seminorms, then E ⊂ X is bounded if and only if for each p ∈ P, sup � p(x) : x ∈ E � < ∞. The dual space of X, abreviated as the dual of X, is the linear space X∗ of all continuous linear functionals x∗ : X → R.1 To begin with, X∗ is just a linear space with no topology. However, two locally convex topologies in X∗ are easy to introduce: • the weak* topology W∗ defined by seminorms x∗ �→ � ��x∗ , x� � � : X∗ → R where x ∈ X; • the strong topology S∗ defined by seminorms �x∗ �B := sup �� ��x∗ , x� � � : x ∈ B � where x∗ ∈ X∗ and B ⊂ X is a bounded set. Since each singleton {x} ⊂ X is a bounded set, the weak* topology is smaller than the strong topology. 1.3. Measures A measure 2 in an arbitrary set X is a function µ defined on all subsets of X that satisfies the following conditions: 1A notable exception to the notation X∗ is the space D� of distributions defined in Section 3.1 below. 2Our concept of measure is often called “outer measure”, and the term “measure” is reserved for the restriction of “outer measure” to the family of all measurable sets. For our purposes, such distinction is superfluous. © 2012 by Taylor & Francis Group, LLC
- 27. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 10 — #20 ✐ ✐ ✐ ✐ ✐ ✐ 10 1. Preliminaries (i) µ(∅) = 0; (ii) µ(B) ≤ µ(A) whenever B ⊂ A ⊂ X; (iii) µ( �∞ k=1 Ak) ≤ �∞ k=1 µ(Ak) whenever Ak ⊂ X for k = 1, 2, . . . . Throughout this section, µ is a measure in a set X ⊂ Rm . The reduction of µ to a set Y ⊂ X is a measure µ Y in X defined by (µ Y )(A) := µ(A ∩ Y ) for each A ⊂ X. If µ = µ Y , we say that µ lives in Y . A set E ⊂ X is called µ measurable whenever µ = µ E + µ (X − E). The support of µ is the set spt µ := X − �� U ⊂ X : U is open in X and µ(U) = 0 � . Since each subset of Rm has the Lindelöf property [28, Section 3.8 and Corol- lary 4.1.16], we have µ(X − spt µ) = 0. In accordance with the standard terminology, the measure µ is called • σ-finite if X = �∞ k=1 Ek and µ(Ek) < ∞ for k = 1, 2, . . . , • Borel if each relatively Borel subset of X is µ measurable, • Borel regular if µ is a Borel measure and each E ⊂ X is contained in a relatively Borel subset B of X such that µ(B) = µ(E), • Radon if µ is a Borel regular measure and µ(K) < ∞ for each compact set K ⊂ X, • metric if µ(A ∪ B) ≥ µ(A) + µ(B) for each pair A, B ⊂ X such that dist (A, B) > 0. A set E ⊂ X is called µ σ-finite if the reduced measure µ E is σ-finite. The next two theorems are proved in [29, Sections 1.1 and 1.9]. Theorem 1.3.1. Let Ω ⊂ Rm be an open set. Each metric measure in Ω is a Borel measure. If µ is a Borel regular measure in Ω, then µ E is a Radon measure for each µ measurable set E ⊂ Ω with µ(E) < ∞. If µ is a Radon measure in Ω, then the following conditions hold: (1) For each set A ⊂ Ω, µ(A) = inf � µ(U) : U ⊂ Ω is open and A ⊂ U � . (2) For each µ measurable set A ⊂ Ω, µ(A) = sup � µ(K) : K ⊂ A is compact � . A set E ⊂ X with µ(E) = 0 is called µ negligible. Sets A, B ⊂ X are µ equivalent if µ(A � B) = 0; they are µ overlapping if µ(A ∩ B) > 0. Maps φ and ψ from a set E ⊂ X to a set Y are µ equivalent if the set {φ �= ψ} := � x ∈ E : φ(x) �= ψ(x) � © 2012 by Taylor & Francis Group, LLC
- 28. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 11 — #21 ✐ ✐ ✐ ✐ ✐ ✐ 1.3. Measures 11 is µ negligible. When the measure µ is clearly understood from the context, we indicate the equivalence by symbols A ∼ B and φ ∼ ψ. Let E ⊂ X be a µ measurable set. A map φ : E → Rs is called µ measurable if the set φ−1 (B) is µ measurable for every Borel set B ⊂ Rs . The linear space of all µ measurable maps φ : E → Rs is denoted by L0 (E, µ; Rs ). The essential support of φ ∈ L0 (E, µ; Rs ) is the set ess spt φ := spt � µ {φ �= 0} � . Unlike the support of φ, the essential support of φ depends only on the µ equivalence class of φ. If φ ∈ L0 (E, µ; Rs ), we also define ess sup x∈E � �φ(x) � � := inf � sup x∈E � �ψ(x) � � : ψ ∈ L0 (E, µ; Rs ) and ψ ∼ φ � . For each φ ∈ L0 (E, µ; Rs ), there is ψ ∼ φ such that ess spt φ = spt ψ and ess sup x∈E � �φ(x) � � = sup x∈E � �ψ(x) � �. Convention 1.3.2. As is customary, we do not explicitly distinguish between an individual set E ⊂ X, or an individual map φ, and the µ equivalence class determined by E, or by φ, respectively. However, the reader should be aware of the following custom: we think of the space L0 (E, µ; Rs ) as consisting of equivalence classes, but when we write φ ∈ L0 (E, µ; Rs ), we view φ as a specific representative of its equivalence class. In particular, when writing φ ∈ L0 (E, µ; Rs ), we always assume that ess spt φ = spt φ and ess sup x∈E � �φ(x) � � = sup x∈E � �φ(x) � �. The next two theorems are standard tools of measure theory. Their proofs can be found in [29, Section 1.2]. Theorem 1.3.3 (Egoroff). Let µ be a finite measure in X ⊂ Rm , and let {φk} be a sequence in L0 (X, µ; Rs ) that converges pointwise. Given ε > 0, there is a µ measurable set E ⊂ X such that µ(X − E) < ε and the sequence {φk � E} converges uniformly. Theorem 1.3.4 (Luzin). Let µ be a finite Borel regular measure in X ⊂ Rm , and let φ ∈ L0 (X, µ; Rs ). Given ε > 0, there is a compact set K ⊂ X such that µ(X − K) < ε and the restriction φ � K is continuous. Given a µ measurable set E ⊂ X, we let �φ�Lp(E,µ;Rs) : = �� E |φ|p dµ �1/p if 1 ≤ p < ∞, �φ�L∞(E,µ;Rs) : = ess sup x∈E � �φ(x) � � © 2012 by Taylor & Francis Group, LLC
- 29. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 12 — #22 ✐ ✐ ✐ ✐ ✐ ✐ 12 1. Preliminaries for each φ ∈ L0 (E, µ; Rs ), and for 1 ≤ p ≤ ∞, define Lp (E, µ; Rs ) : = � f ∈ L0 (E, µ; Rs ) : �f�Lp(E,µ;Rs) < ∞ � . Let 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞ be such that 1/p + 1/q = 1 where we define 1/∞ := 0. The Hölder inequality �fg�L1(E,µ;Rs) ≤ �f�Lp(E,µ;Rs)�g�Lq(E,µ;Rs) (1.3.1) holds for each f, g ∈ L0 (E, µ; Rs ); see [63, Theorem 3.5]. Let µ be a Borel measure in an open setΩ ⊂ Rm . For 1 ≤ p ≤ ∞, we denote by Lp loc(Ω, µ; Rs ) the linear space of all maps φ ∈ L0 (Ω, µ; Rs ) such that φ � U belongs to Lp (U, µ; Rs ) for each open set U � Ω. Unless stated otherwise, throughout we assume that Lp loc(Ω, µ; Rs ) has been equipped with the Fréchet topology defined by seminorms φ �→ �φ � U�Lp(U,µ;Rs) : Lp loc(Ω, µ; Rs ) → R where U � Ω is an open set; cf. Example 1.2.2. We write Lp (E, µ) and Lp loc(Ω, µ) instead of Lp (E, µ; R) and Lp loc(Ω, µ; R), respectively. The following theorem is essential for establishing Theorem 2.3.7 and Proposition 7.4.3 below; in addition, it simplifies proofs of some differentiation results (Theorems 4.3.4 and 6.2.3 below). We call it the Henstock lemma, but the name Saks-Henstock lemma is also used — cf. [44] and [37]. Theorem 1.3.5. Let µ be a Radon measure in X ⊂ Rm , let E ⊂ X be a µ measurable set with µ(E) < ∞, and let f ∈ L1 (E, µ) be real-valued. Given ε > 0, there is δ : E → R+ satisfying the following condition: for every collection {E1, . . . , Ep} of µ measurable µ nonoverlapping subsets of E, and for every set of points {x1, . . . , xp} ⊂ E, the inequality p � i=1 � � � �f(xi)µ(Ei) − � Ei f dµ � � � � < ε holds whenever d � Ei ∪ {xi} � < δ(xi) for i = 1, . . . , p. Proof. Choose ε > 0, and using the Vitali-Carathéodory theorem [63, The- orem 2.25], find functions g and h defined on E that are, respectively, upper and lower semicontinuous, and satisfy g ≤ f ≤ h and � E (h − g) dµ < ε. There is δ : E → R+ such that g(y) < f(x) + ε and h(y) > f(x) − ε for all x, y ∈ E with |x − y| < δ(x). If {E1, . . . , Ep} and {x1, . . . , xp} satisfy the © 2012 by Taylor & Francis Group, LLC
- 30. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 13 — #23 ✐ ✐ ✐ ✐ ✐ ✐ 1.4. Hausdorff measures 13 conditions of the proposition, then � Ei g dµ ≤ � Ei f dµ ≤ � Ei h dµ, � Ei g dµ − εµ(Ei) ≤ f(xi)µ(Ei) ≤ � Ei h dµ + εµ(Ei) for i = 1, . . . , p. Consequently p � i=1 � � � �f(xi)µ(Ei) − � Ei f dµ � � � � ≤ p � i=1 �� Ei (h − g) dµ + εµ(Ei) � ≤ � E (h − g) dµ + εµ(E) < ε � 1 + µ(E) � . Lebesgue measure in Rm is denoted by Lm . For each subset E of the ambient space Rn , we let |E| := Ln (E). Sets A, B ⊂ Rn are called overlapping if they are Ln overlapping, that is to say if |A ∩ B| > 0. Unless specified otherwise, all concepts connected with measures, such as “measurable”, “negligible”, etc., as well as the expressions “almost all” and “almost everywhere”, refer to the measure Ln in Rn . For a measurable set E ⊂ Rn , we let Lp (E; Rs ) := Lp (E, Ln ; Rs ) and Lp (E) := Lp (E, Ln ). IfΩ ⊂ Rn is an open set, the meanings of Lp loc(Ω; Rs ) and Lp loc(Ω) are obvious. When no confusion is possible, we write � E f(x) dx or � E f instead of � E f dLn . 1.4. Hausdorff measures We define Hausdorff measures in Rn , and state some of their elementary properties. Select a fixed s ≥ 0, and let Γ(s) := � ∞ 0 ts−1 e−t dt and α(s) := Γ �1 2 �s Γ �s 2 + 1 �. Recall that Γ : t �→ Γ(t) is the classical Euler’s gamma function [62, Defini- tion 8.17]. Using Fubini’s theorem and induction, we obtain α(n) = Ln �� x ∈ Rn : |x| ≤ 1 �� ; a more advanced calculation can be found in [56, Chapter 1, Equation 1.1.7]. The function α : t �→ α(t) maps [0, ∞) to [1, 5), has only one local maximum © 2012 by Taylor & Francis Group, LLC
- 31. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 14 — #24 ✐ ✐ ✐ ✐ ✐ ✐ 14 1. Preliminaries and one local minimum, attained at 5 < tmax < 6 and tmin = 0, respectively; in addition α(t) → 0 as t → ∞. For E ⊂ Rn and δ > 0, let Hs δ(E) := inf ∞ � k=1 α(s) � d(Ck) 2 �s (1.4.1) where the infimum is taken over all sequences {Ck} of subsets of Rn such that E ⊂ �∞ k=1 Ck and d(Ck) < δ for k = 1, 2, . . . ; here we define 00 := 1 and d(∅)s := 0. Letting Hs (E) := sup δ>0 Hs δ(E) = lim δ→0 Hs δ(E), the function Hs : E �→ Hs (E), defined for each E ⊂ Rn , is a measure in Rn , called the s-dimensional Hausdorff measure. Since Hs (A ∪ B) = Hs (A) + Hs (B) for every pair of sets A, B ⊂ Rn with dist(A, B) > 0, it follows from The- orem 1.3.1 that Hs is a Borel measure in Rn . In addition, the measure Hs is Borel regular by Proposition 1.4.2 below. However, Hs is not a Radon measure in Rn when s < n. It is easy to verify that H0 is the counting measure in Rn . The constant α(s)/2s in the definition of Hs δ(E) implies Hn = Ln . This equality follows (nontrivially) from the isodiametric inequality Ln (E) ≤ α(n) � d(E) 2 �n (1.4.2) which holds for every E ⊂ Rn ; see [29, Section 2.2]. As the diameters of sets are invariant with respect to isometric transformations, so are the Hausdorff measures. Moreover, for each E ⊂ Rn and every t > 0, Hs (tE) = ts Hs (E). Remark 1.4.1. The value of Hs δ(E), and a fortiori that of Hs (E), does not change when the sequence {Ck} in the defining equality (1.4.1) is assumed to have one of the following additional properties: (1) Each Ck is convex; since the diameters of Ck and its convex hull are the same. (2) Each Ck is closed; since the diameters of Ck and its closure cl Ck are the same. (3) Each Ck is open; since given ε > 0, we can find rk > 0 such that d � U(Ck, rk) � < δ and d � U(Ck, rk) �s < d(Ck)s + ε2−k for k = 1, 2, . . . . (4) Each Ck is contained in E; since E is covered by the family {Ck ∩ E} and d(Ck ∩ E) ≤ d(Ck) for k = 1, 2, . . . . © 2012 by Taylor & Francis Group, LLC
- 32. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 15 — #25 ✐ ✐ ✐ ✐ ✐ ✐ 1.4. Hausdorff measures 15 By (4), the value Hs (E) depends on Rn only to the extent to which Rn defines the metric in E. In particular, if 1 ≤ m < n is an integer, then Hs restricted to the subsets of Rm is the s-dimensional Hausdorff measure in Rm . Proposition 1.4.2. Given E ⊂ Rn , there is a Gδ set B ⊂ Rn such that E ⊂ B and Hs (E) = Hs (B). In particular, Hs is a Borel regular measure. Proof. Assume Hs (E) < ∞, since otherwise it suffices to let B := Rn . Fix k ∈ N. By Remark 1.4.1, (3), there are open sets Uk,j ⊂ Rn such that d(Uk,j) < 1/k for j = 1, 2, . . . , Uk = �∞ j=1 Uk,j contains E, and Hs 1/k(Uk) ≤ ∞ � j=1 α(s) � d(Uk,j) 2 �s < Hs (E) + 1 k . The first inequality follows directly from the definition of Hs 1/k. The intersec- tion B = �∞ k=1 Uk is a Gδ set containing E, and Hs 1/k(B) ≤ Hs 1/k(Uk) < Hs (E) + 1 k . Letting k → ∞ yields Hs (B) ≤ Hs (E). Proposition 1.4.3. Let E ⊂ Rn , and let 0 ≤ s < t. If the measure Hs E is σ-finite, then Ht (E) = 0. Moreover, Hs ≡ 0 for each s > n. Proof. In proving the first claim, we may assume that Hs (E) < ∞. Given δ > 0, there is a sequence {Ck} of subsets of Rn of diameters smaller than δ such that E ⊂ �∞ k=1 Ck and ∞ � k=1 α(s) � d(Ck) 2 �s < Hs (E) + 1. Consequently Ht δ(E) ≤ ∞ � k=1 α(t) � d(Ck) 2 �t ≤ � δ 2 �t−s α(t) α(s) ∞ � k=1 α(s) � d(Ck) 2 �s ≤ � δ 2 �t−s α(t) α(s) � Hs (E) + 1 � , and it suffices to let δ → 0. If s > n, it suffices to show that Hs (Q) = 0 for a 0-cube Q. For k ∈ N, each k-cube has diameter δk := 2−k √ n, and Q is the union of 2kn such cubes. Thus Hs δk (Q) ≤ α(s)2kn � δk 2 �s = α(s) �√ n 2 �s 2k(n−s) , and letting k → ∞ yields the desired result. Next we relate Hausdorff measures in Rn to covers consisting of dyadic cubes. Recall that for k ∈ Z, the family of all k-cubes is denoted by Dk. © 2012 by Taylor & Francis Group, LLC
- 33. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 16 — #26 ✐ ✐ ✐ ✐ ✐ ✐ 16 1. Preliminaries Proposition 1.4.4. Let s ≥ 0 and E ⊂ Rn . Given ε > 0 and p ∈ Z, there is a family Q ⊂ � k≥p Dk such that E meets each Q ∈ Q, E ⊂ int �� Q � , and � Q∈Q d(Q)s ≤ β � Hs (E) + ε � where β = β(n) > 0. Proof. If s > n, the proposition holds with β = 1; the proof is analogous to that of the second part of Proposition 1.4.3. Hence assume s ≤ n, and choose ε > 0 and p ∈ Z. There is a cover {Cj} of E such that the diameter of each Cj is smaller than δ = 2−p , and ∞ � j=1 α(s) � d(Cj) 2 �s ≤ Hs δ(E) + ε ≤ Hs (E) + ε. Find an integer pj ≥ p with 2−pj −1 ≤ d(Cj) < 2−pj , and note d(Cj) < d(Q)/ √ n = 2−pj ≤ 2d(Cj) for every pj-cube Q. Select a pj-cube Q with Q ∩ Cj �= ∅, and denote by Q1,j, . . . , Q3n,j all pj-cubes which meet Q, including Q itself. It follows that Cj ⊂ int ��3n i=1 Qi,j � , and hence E ⊂ int ��∞ j=1 �3n i=1 Qi,j � . Moreover, ∞ � j=1 3n � i=1 d(Qi,j)s ≤ 3n · � 2 √ n �s ∞ � j=1 d(Cj)s ≤ 3n (4n)s α(s) � Hs (E) + ε � . Since 0 ≤ s ≤ n implies α(s) ≥ min � 1, α(n) � , the desired inequality holds with β := (12n)n max � 1, 1/α(n) � . Finally, replacing Q by a smaller family {Q ∈ Q : Q ∩ E �= ∅} completes the proof. Additional properties of Hausdorff measures in Rn can be found in [30] and [46]. Hausdorff measures defined in general metric spaces are investigated in [59]. 1.5. Differentiable and Lipschitz maps LetΩ ⊂ Rn be an open set. A map φ : Ω → Rm is differentiable at x ∈ Ω if there is a linear map L : Rn → Rm such that lim y→x � �φ(y) − φ(x) − L(y − x) � � |y − x| = 0. If such a map L exists, it is unique. We call it the derivative of φ at x, denoted by Dφ(x). If a map φ = (f1, . . . , fn) from Ω to Rn is differentiable at x ∈ Ω, © 2012 by Taylor & Francis Group, LLC
- 34. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 17 — #27 ✐ ✐ ✐ ✐ ✐ ✐ 1.5. Differentiable and Lipschitz maps 17 then the divergence of φ at x is the real number div φ(x) := n � i=1 Difi(x) where Di := ∂/∂ξi is the usual partial derivative operator. Let E ⊂ Rn be any set, and let φ : E → Rm . The Lipschitz constant of φ is the extended real number Lip φ := sup �� �φ(x) − φ(y) � � |x − y| : x, y ∈ E and x �= y � . When Lip φ < ∞, the map φ is called Lipschitz. If Ω⊂ Rn is an open set, we call a map φ : Ω → Rm locally Lipschitz whenever the restric- tion φ � U is Lipschitz for each open set U � Ω. The linear space of all Lipschitz maps φ : E → Rm is denoted by Lip(E; Rm ). The symbols Lip(E), Lipc(Ω; Rm ), Liploc(Ω; Rm ), etc., have the obvious meaning. For a Lipschitz map φ : E → Rm and s ≥ 0, we obtain Hs � φ(E) � ≤ (Lip φ)s Hs (E). (1.5.1) A bijective Lipschitz map whose inverse is also Lipschitz is called a lipeomor- phism. Observation 1.5.1. Let Ω ⊂ Rn be an open set. If φ ∈ Liploc(Ω; Rm ), then φ � K ∈ Lip(C; Rm ) for each compact set K ⊂ Ω. In particular, Liploc(Ω; Rm ) ∩ Cc(Ω; Rm ) = Lipc(Ω; Rm ). Proof. Suppose there is a compact set K ⊂ Ω such that φ is not Lipschitz in K. There are sequences {xk} and {yk} in K such that � �φ(xk) − φ(yk) � � > k|xk − yk| > 0, k = 1, 2, . . . . Passing to subsequences, still denoted by {xk} and {yk}, we obtain the limit points x = lim xk and y = lim yk in K. The continuity of φ implies ∞ > � �φ(x) − φ(y) � � = lim � �φ(xk) − φ(yk) � � ≥ lim sup k|xk − yk|, and consequently x = y. Since φ is Lipschitz in a neighborhood of x, there is 0 < c < ∞ such that for all sufficiently large k, c|xk − yk| ≥ � �φ(xk) − φ(yk) � � > k|xk − yk| > 0. A contradiction follows. Proposition 1.5.2. Let E ⊂ Rn and φ ∈ Lip(E; Rm ). There is a map ψ ∈ Lip(Rn ; Rm ) such that ψ(x) = φ(x) for each x ∈ E, Lip ψ ≤ √ m Lip φ, and �ψ�L∞(Rn;Rm) ≤ �φ�L∞(E;Rm). © 2012 by Taylor & Francis Group, LLC
- 35. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 18 — #28 ✐ ✐ ✐ ✐ ✐ ✐ 18 1. Preliminaries Proof. By [29, Section 3.1.1, Theorem 1], there is θ ∈ Lip(Rn ; Rm ) such that θ(x) = φ(x) for each x ∈ E and Lip θ ≤ √ m Lip φ. As there is nothing to prove otherwise, assume c := �φ�L∞(E;Rm) belongs to R+. Define γ : Rm → Rm by γ(y) := � c y |y| if |y| > c, y if |y| ≤ c. Since Lip γ = 1 and |γ|L∞(Rm;Rm) ≤ c, the composition ψ := γ ◦ θ is the desired extension of φ. With a considerable effort, one can improve on Proposition 1.5.2 by show- ing that Lip ψ = Lip φ. This stronger result is called Kirschbraun’s theorem [33, Theorem 2.10.43]. The next well-known theorem has several proofs of various levels of so- phistication, e.g., [33, Theorem 3.1.6], [1, Theorem 2.14], or [29, Section 6.2, Theorem 2]. For a proof with minimal prerequisites we refer to [29, Sec- tion 3.1.2]. Theorem 1.5.3 (Rademacher). Each φ ∈ Lip(Rn ; Rm ) is differentiable at almost all x ∈ Rn . Let E ⊂ Rn and 0 ≤ s ≤ 1. The s-Hölder constant at x ∈ E of a map φ : E → Rm is the extended real number Hsφ(x) := lim sup y→x y∈E � �φ(y) − φ(x) � � |y − x|s . Clearly, H0φ(x) < ∞ if and only if φ is bounded in a neighborhood of x, and H0φ(x) = 0 if and only if φ is continuous at x. If Hsφ(x) < ∞ and 0 ≤ t < s, then Htφ(x) = 0. We call Lip φ(x) := H1φ(x) the Lipschitz constant of φ at x, and say that φ is Lipschitz at x whenever Lip φ(x) < ∞. We say that φ is pointwise Lipschitz in a set C ⊂ E if it is Lipschitz at each x ∈ C. A pointwise Lipschitz map in C need not be Lipschitz in C, even if C is compact [51, Section 1.6]. Theorem 1.5.4 (Stepanoff). Let Ω ⊂ Rn be an open set, and assume that φ : Ω → Rm is pointwise Lipschitz in a set E ⊂ Ω. Then φ is differentiable at almost all x ∈ E. For a proof of this generalization of Rademacher’s theorem we refer to [33, Theorem 3.1.9], or to Section 6.2 below where a slightly more general theorem is proved in detail; see Remark 7.2.4. © 2012 by Taylor & Francis Group, LLC
- 36. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 19 — #29 ✐ ✐ ✐ ✐ ✐ ✐ 1.5. Differentiable and Lipschitz maps 19 Theorem 1.5.5 (Whitney). Let K ⊂ Rn be a compact set, and let f ∈ C(K) and v ∈ C(K; Rn ) satisfy the following condition: given ε > 0, we can find δ > 0 so that � �f(y) − f(x) − v(x) · (y − x) � � ≤ ε|y − x| for all x, y ∈ K with |y − x| < δ. There is g ∈ C1 (Rn ) such that g(x) = f(x) and Dg(x) = v(x) for each x ∈ K. Theorem 1.5.5 is a special case of Whitney’s extension theorem. Proofs of the general Whitney’s result, which implies the special case, can be found in [29, Section 6.5] or in [70, Chapter 6, Section 2]. Let φ = (f1, . . . , fn) be a Lipschitz map from Rn to Rn . Then Df1 . . . Dfn is an n × n matrix, whose determinant is denoted by det Dφ. The Jaco- bian of φ is the function Jφ = | det Dφ| defined almost everywhere in Rn by Rademacher’s theorem. In view of (1.5.1), the inequality �Jφ�L∞(Rn) ≤ (Lip φ)n (1.5.2) is a consequence of [29, Section 3.3, Lemma 1]. The next result is called interchangeably the area theorem or change of variables theorem. It follows from [29, Section 3.3, Theorem 2]. Theorem 1.5.6. Let φ : Rn → Rn be a Lipschitz map. If g ∈ L0 (Rn ) and g ≥ 0, then y �→ �� g(x) : x ∈ φ−1 (y)} is a measurable function on Rn and � Rn g(x)Jφ(x) dx = � Rn � x∈φ−1(y) g(x) dy. (1.5.3) Employing Hausdorff measures and more elaborete Jacobians, formulas similar to (1.5.3) hold for Lipschitz maps φ : Rn → Rm where m �= n. On a few occasions when such formulas are used, we refer the reader to the appropriate sections of [29, Chapter 3]. © 2012 by Taylor & Francis Group, LLC
- 38. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 21 — #31 ✐ ✐ ✐ ✐ ✐ ✐ Chapter 2 Divergence theorem for dyadic figures Using the idea of W.B. Jurkat, we give an elementary proof of a fairly general divergence theorem for dyadic figures. While this is only a preliminary version of the divergence theorem we intend to establish, it is already a useful tool for studying removable singularities of some classical partial differential equations (Chapter 3 below). 2.1. Differentiable vector fields If A is a figure, then for Hn−1 almost all x ∈ ∂A there is a unique unit exterior normal of A at x, denoted by νA(x). The map νA : x �→ νA(x) : ∂A → Rn is defined Hn−1 almost everywhere, has only finitely many values, and it is Hn−1 measurable. Let E ⊂ Rn , and assume that v : E → Rn belongs to L1 (∂A, Hn−1 ; Rn ) for each figure A ⊂ E. A real-valued function F : A �→ � ∂A v · νA dHn−1 (2.1.1) defined on all figures A ⊂ E is called the flux of v. In this context it is customary to call v a vector field. The term “flux” is derived from a physical example: if v is the vector field of velocities of a fluid moving in the set E, then F(A) is the amount of fluid that flows out of the figure A ⊂ E in the unit of time. The next observation says that the flux of a vector field is an additive function with respect to nonoverlapping figures. Its simple verification is left to the reader. Observation 2.1.1. Let E ⊂ Rn , and assume that v : E → Rn belongs to L∞ (∂A, Hn−1 ; Rn ) for each figure A ⊂ E. Then � ∂(A∪B) v · νA∪B dHn−1 = � ∂A v · νA dHn−1 + � ∂B v · νB dHn−1 for each pair A, B ⊂ E of nonoverlapping figures. 21 © 2012 by Taylor & Francis Group, LLC
- 39. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 22 — #32 ✐ ✐ ✐ ✐ ✐ ✐ 22 2. Divergence theorem for dyadic figures Proposition 2.1.2. Let A be a cell, and let v ∈ C(A; Rn ) be differentiable at each x ∈ int A. If div v belongs to L1 (A), then � A div v(x) dx = � ∂A v · νA dHn−1 . Proof. Let v = (v1, . . . , vn) and A = �n i=1[a− i , a+ i ]. If A± i := {a± i } × πi(int A), then νA(x) = ±ei whenever x ∈ A± i , i = 1, . . . , n. The boundary ∂A differs from �n i=1(A− i ∪ A+ i ) by an Hn−1 negligible set. Fix i and for x ∈ int A, write x = (u, t) where u = πi(x) and t = x · ei. By Fubini’s theorem and the fundamental theorem of calculus, � A Divi(x) dx = � πi(int A) �� a+ i a− i d dt vi(u, t) dt � du = � πi(int A) � vi(u, a+ i ) − vi(u, a− i ) � du = � A+ i v · νA dHn−1 + � A− i v · νA dHn−1 . Summing up these equalities over i = 1, . . . , n completes the proof. Corollary 2.1.3. Let E ⊂ Rn , x ∈ E, and let {Ck} be a sequence of cubes such that lim d(Ck) = 0. Assume that Ck ⊂ E and x ∈ Ck for k = 1, 2, . . . , and that v : E → Rm belongs to L1 (∂C, Hn−1 ; Rn ) for each cube C ⊂ E. (1) If 0 ≤ s ≤ 1, then lim sup 1 d(Ck)n−1+s � ∂Ck v · νCk dHn−1 ≤ 2nHsv(x). (2) If x ∈ int E and v is differentiable at x, then lim 1 |Ck| � ∂Ck v · νCk dHn−1 = div v(x). Proof. Choose ε > 0 — you can never go wrong by doing so. We may assume Hsv(x) < ∞, and find δ > 0 so that � �v(y) − v(x) � � ≤ � Hsv(x) + ε � · |y − x|s © 2012 by Taylor & Francis Group, LLC
- 40. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 23 — #33 ✐ ✐ ✐ ✐ ✐ ✐ 2.1. Differentiable vector fields 23 for each y ∈ E ∩U(x,δ ). Denote by F the flux of v, and use Proposition 2.1.2 to show that for all sufficiently large k, � �F(Ck) � � = � � � � � ∂Ck � v(y) − v(x) � · νCk (y) dHn−1 (y) � � � � ≤ � Hsv(x) + ε � � ∂Ck |y − x|s dHn−1 (y) ≤ � Hsv(x) + ε � d(Ck)s Hn−1 (∂Ck) ≤ 2n � Hsv(x) + ε � d(Ck)n−1+s . If x ∈ int E and v is differentiable at x, let w : y �→ v(x) + � Dv(x) � (y − x) : Rn → Rn and observe that div w(y) = div v(x) for each y ∈ Rn . There is η > 0 such that U(x,η ) ⊂ E and � �v(y) − w(y) � � ≤ ε|y − x| for every y ∈ U(x,η ). As w ∈ C∞ (Rn ; Rn ), Proposition 2.1.2 yields � � �F(Ck) − div v(x)|Ck| � � � = � � � � � ∂Ck � v(y) − w(y) � · νCk (y) dHn−1 (y) � � � � ≤ ε � ∂Ck |y − x| dHn−1 (y) ≤ εd(Ck)Hn−1 (∂Ck) = 2n3/2 ε|Ck| for all sufficiently large k. Letting k → ∞, the corollary follows from the arbitrariness of ε. We prove the divergence theorem for closed balls. While this is not essential for the logical development of our exposition, it will facilitate an early presentation of examples. If B := B(x, r), then νB(y) := (y − x)/r is the unit exterior normal of B at y ∈ ∂B. Since the induced map νB : ∂B → Rn is continuous, a finite integral � ∂B v · νB dHn−1 exists for each v ∈ L1(∂B, Hn−1; Rn). Proposition 2.1.4. Let B ⊂ Rn be a closed ball, and let v ∈ C(B; Rn) be differentiable in each x ∈ int B. If div v belongs to L1(B), then � B div v(x) dx = � ∂B v · νB dHn−1 . Proof. The proof is similar to that of Proposition 2.1.2. In view of translation invariance, we may assume B = B(0, r). Let U := πn(int B) and g(u) = � r2 − |u|2 for each u ∈ U. If (∂B)± := {x ∈ ∂B : ±x · en > 0}, then the bijections φ± : u �→ � u, ±g(u) � : U → (∂B)± are continuously differentiable and have the same Jacobian J = � 1 + � �Dg � �2 = r g ; © 2012 by Taylor & Francis Group, LLC
- 41. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 24 — #34 ✐ ✐ ✐ ✐ ✐ ✐ 24 2. Divergence theorem for dyadic figures see [29, Section 3.3.4, B]. Let vn := v · en and νn := νB · en. Observe νn ◦ φ± = (νB ◦ φ±) · en = φ± · en r = ± g r . If Uk = πn � U(0, 1 − 2−k) � , then the maps φ± � Uk are Lipschitz and U = �∞ k=1 Uk. Thus applying [29, Section 3.3.4, B] to each Uk, we obtain � (∂B)± vnνn dHn−1 = � U � (vnνn) ◦ φ± � J dLn = ± � U vn � u, ±g(u) � du. Now Fubini’s theorem and the fundamental theorem of calculus imply � B Dnvn(x) dx = � U �� g(u) −g(u) d dt vn(u, t) dt � du = � U vn � u, g(u) � du − � U vn � u, −g(u) � du = � (∂B)+ vnνn dHn−1 + � (∂B)− vnνn dHn−1 = � ∂B vnνn dHn−1 , since the boundary ∂B differs from (∂B)+ ∪ (∂B)− by an Hn−1 negligible set. The propo- sition follows from symmetry. 2.2. Dyadic partitions A partition is a finite (possibly empty) collection P := � (E1, x1), . . . , (Ep, xp) � where {E1, . . . , Ep} is a collection of nonoverlapping subsets of Rn such that xi ∈ Ei for i = 1, . . . , p. The body of P is the union [P] := �p i=1 Ei, and P is called a partition in a set A ⊂ Rn if [P] ⊂ A. Given a set E ⊂ Rn and δ : E → R+, we say that P is δ-fine if xi ∈ E and d(Ei) < δ(xi) for i = 1, . . . , p. When each set Ei is a dyadic cube, then P is called a dyadic partition. If dyadic cubes A and B overlap, then either A ⊂ B or B ⊂ A. Conse- quently, every family C of dyadic cubes has a nonoverlapping subfamily Q such that � Q = � C. Dyadic cubes A and B are called adjacent if d(A) = d(B) and A ∩ B �= ∅. Every dyadic cube is adjacent to 3n dyadic cubes, including itself. Recall that for an integer k the family of all k-cubes is denoted by Dk. Given a family E of subsets of Rn and x ∈ Rn , we let St(x, E) := {E ∈ E : x ∈ E}. For each x ∈ Rn and each k ∈ Z, the collection St(x, Dk) consists of at most 2n k-cubes, and x belongs to the interior of � St(x, Dk). A star cover of © 2012 by Taylor & Francis Group, LLC
- 42. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 25 — #35 ✐ ✐ ✐ ✐ ✐ ✐ 2.2. Dyadic partitions 25 E ⊂ Rn is a family Q of dyadic cubes such that for each x ∈ E there is kx ∈ Z with St(x, Dkx ) ⊂ Q; in this case E ⊂ � x∈E int �� St(x, Dkx ) � ⊂ int �� Q � . It follows that a star cover Q of a compact set K ⊂ Rn has a finite nonover- lapping subcover. Lemma 2.2.1. Let δ be a positive function defined on a set E ⊂ Rn , and let 0 ≤ t ≤ n. Given ε > 0, the set E has a star cover C which satisfies the following conditions: (1) for each C ∈ C there is xC ∈ C ∩ E such that d(C) < δ(xC); (2) � C∈C d(C)t ≤ κ � Ht (E) + ε � where κ := κ(n) > 0. Proof. To avoid trivialities, assume E �= ∅. Denote by B the family of all dyadic cubes C satisfying condition (1). For k ∈ N and x ∈ Rn , let D≥k := � {Di : i ≥ k} and Bk := � x ∈ Rn : St(x, D≥k) ⊂ B � . Clearly {Bk} is an increasing sequence. Moreover E ⊂ �∞ k=1 Bk, since � x ∈ E : δ(x) > 2−k √ n � ⊂ Bk. Claim. Rn − Bk = � (D≥k − B) for every k ∈ N. In particular, each Bk is a Borel set. Proof . If x �∈ Bk, some Cx ∈ St(x, D≥k) does not belong to B. Hence x ∈ Cx and Cx ∈ D≥k − B. It follows that x ∈ � (D≥k − B). Conversely, if x ∈ � (D≥k−B) then x ∈ Dx for some Dx ∈ D≥k−B. Thus St(x, D≥k) �⊂ B, which means x �∈ Bk. If E1 := E ∩ B1 and Ek := E ∩ (Bk − Bk−1) for k = 2, 3, . . . , then E = ∞ � k=1 Ek and Ht (E) = ∞ � k=1 Ht (Ek). Select Ek �= ∅. By Proposition 1.4.4, there is a cover Qk ⊂ D≥k of Ek such that Ek ∩ Q �= ∅ for each Q ∈ Qk, and � Q∈Qk d(Q)t ≤ β � Ht (Ek) + ε2−k � where β = β(n) > 0. If Ck consists of all dyadic cubes that meet Ek and are adjacent to some Q ∈ Qk, then Ck is a star cover of Ek, and � C∈Ck d(C)t ≤ 3n � Q∈Qk d(Q)t ≤ 3n β � Ht (Ek) + ε2−k � . © 2012 by Taylor & Francis Group, LLC
- 43. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 26 — #36 ✐ ✐ ✐ ✐ ✐ ✐ 26 2. Divergence theorem for dyadic figures Now Ck ⊂ D≥k and Ek ⊂ Bk. Since each C ∈ Ck meets Ek, the definition of Bk implies C ∈ B. Thus Ck ⊂ B, and we see that the family C := �∞ k=1 Ck is a star cover of E that satisfies condition (1). Letting κ := 3n β, we obtain � C∈C d(C)t ≤ ∞ � k=1 � C∈Ck d(C)t ≤ κ ∞ � k=1 � Ht (Ek) + ε2−k � = κ � Ht (E) + ε � . Proposition 2.2.2. Let E be a family of disjoint subsets of a dyadic figure A, and for each E ∈ E select real numbers 0 ≤ tE ≤ n and εE > 0. Given δ : A → R+, there is a δ-fine dyadic partition P := � (C1, x1), . . . , (Cp, xp) � such that [P] = A, and with a fixed κ = κ(n) > 0, the inequality � xi∈E d(Ci)tE ≤ κ � HtE (E) + εE � holds for each E ∈ E. Proof. There is k ∈ N such that A is the union of k-cubes. Enlarging E and making δ smaller, we may assume � E = A and δ(x) < 2−k √ n for each x ∈ A. Let CE be a star cover of E ∈ E associated with δE := δ � E, tE, and εE according to Lemma 2.2.1. For every C ∈ CE, select xC ∈ E ∩ C with d(C) < δE(xC). Since C := � E∈E CE is a star cover of the compact set A, there are nonoverlapping cubes C1, . . . , Cp in C such that A ⊂ �p i=1 Ci. It follows that P := � (Ci, xCi ) : |Ci ∩ A| > 0 � is a δ-fine dyadic partition with A ⊂ [P]. As our assumption about δ implies Ci ⊂ A whenever |Ci ∩ A| > 0, we obtain [P] = A. Since E is a disjoint family, {Ci : xCi ∈ E} ⊂ CE for each E ∈ E. Hence with the same κ as in Lemma 2.2.1, the inequality � xCi ∈E d(Ci)tE ≤ � C∈CE d(C)tE ≤ κ � HtE (E) + εE � holds for every E ∈ E. Remark 2.2.3. Lemma 2.2.1 and Proposition 2.2.2 are due to W.B. Jurkat [42, Section 4]. The classical Cousin’s lemma [17] or [51, Lemma 2.6.1], as well as its generalization obtained by E.J. Howard [41, Lemma 5], are immediate consequences of Proposition 2.2.2. © 2012 by Taylor & Francis Group, LLC
- 44. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 27 — #37 ✐ ✐ ✐ ✐ ✐ ✐ 2.3. Admissible maps 27 2.3. Admissible maps Definition 2.3.1. Let E ⊂ Rn be any set. A map φ : E → Rm is called admissible if there are numbers 0 ≤ sk < 1, and disjoint, possibly empty, sets Ek ⊂ E such that φ is pointwise Lipschitz in E− �∞ k=1 Ek and for k = 1, 2, . . . , the following conditions hold: (i) Ek is Hn−1+sk σ-finite, and Hsk φ(x) < ∞ for each x ∈ Ek; (ii) Hn−1+sk (Ek) > 0 implies Hsk φ(x) = 0 for each x ∈ Ek. The family of all admissible maps from the set E to Rm is denoted by Adm(E; Rm ), and we write Adm(E) instead of Adm(E; R). Note that in Definition 2.3.1 no topological restrictions are placed on the exceptional sets Ek. Remark 2.3.2. If Hn−1+sk (Ek) = ∞, then Ek = � j∈N Ek,j where Ek,j are disjoint sets with Hn−1+sk (Ek,j) < ∞. Thus replacing each pair (Ek, sk) with Hn−1+sk (Ek) = ∞ by the collection � (Ek,j, sk) : j ∈ N � , condition (i) of Definition 2.3.1 can be replaced by the condition: (i*) Hn−1+sk (Ek) < ∞, and Hsk φ(x) < ∞ for each x ∈ Ek. This observation will simplify future arguments. Remark 2.3.3. Let E ⊂ Rn and φ ∈ Adm(E; Rm ). Then H0φ(x) < ∞ for all x ∈ E, and H0φ(x) = 0 for all x ∈ E − T where T ⊂ E is Hn−1 negligible. Thus φ is locally bounded in E and continuous in E −T. It follows that if E is Hn−1 measurable then so is φ, and if E is compact then φ is bounded. Since each set Ek is negligible, Stepanoff’s theorem implies that φ is differentiable at almost all x ∈ int E. The restriction φ � B belongs to Adm(B; Rm ) for each B ⊂ A. Remark 2.3.4. A commonly encountered map φ : E → Rm is locally bounded in E, continuous outside an Hn−1 negligible set T ⊂ E, and point- wise Lipschitz outside an Hn−1 σ-finite set S ⊂ E [53, Theorem 2.9]. Letting sk := 0 for k = 1, 2, . . . , E1 := T, E2 := S − T, and Ek = ∅ for k = 3, 4, . . . , we see that φ is admissible. Since locally bounded and pointwise Lipschitz are extreme points of the scale represented by Hölder constants, considering admissible maps is natural. Proposition 2.3.5. Let E ⊂ Rn . With respect to pointwise addition and multiplication, Adm(E) is a commutative ring, and Adm(A; Rm ) is a module over Adm(E). © 2012 by Taylor & Francis Group, LLC
- 45. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 28 — #38 ✐ ✐ ✐ ✐ ✐ ✐ 28 2. Divergence theorem for dyadic figures Proof. Choosing φ,ψ ∈ Adm(E; Rm ) and g ∈ Adm(E), it suffices to show that φ + ψ and gφ belong to Adm(E; Rm ). As the other proof is similar, we show only that θ = gφ belongs to Adm(E; Rm ). Let {rk} and {sk} be sequences in [0, 1), and {Ak} and {Bk} be sequences of disjoint subsets of E, associated with φ and g, respectively, according to Definition 2.3.1. Further let A0 := E − � k∈N Ak, B0 := E − � k∈N Bk, and r0 = s0 = 1. Define ti,j := min{ri, sj} and Ei,j := Ai ∩ Bj for i, j = 0, 1, . . . , and observe that 0 ≤ ti,j < 1 whenever (i, j) �= (0, 0), and that E is the union of a disjoint collection {Ei,j : i, j = 0, 1, . . . }. By Remark 2.3.3, both φ and g are locally bounded in E. A direct calculation shows that there are functions a, b : E → R+ such that Hti,j θ(x) ≤ a(x)Hri φ(x) + b(x)Hsj g(x) for each x ∈ Ei,j and i, j = 0, 1, . . . . Thus θ is pointwise Lipschitz in E0,0 = E − �� Ei,j : i, j = 0, 1, . . . and (i, j) �= (0, 0) � , and for each pair (i, j) �= (0, 0), the following conditions hold: (i) Ei,j is Hn−1+ti,j σ-finite, and Hti,j θ(x) < ∞ for each x ∈ Ei,j; (ii) Hn−1+ti,j (Ei,j) > 0 implies Hti,j θ(x) = 0 for each x ∈ Ei,j. This verifies that θ is an admissible map. Lemma 2.3.6. Let A be a dyadic figure, let v ∈ Adm(A; Rn ), and define f : A → R by the formula f(x) := � div v(x) if x ∈ int A and v is differentiable at x, 0 otherwise. For each ε > 0 and each δ : A → R+, there is a δ-fine dyadic partition P := � (C1, x1), . . . , (Cp, xp) � such that [P] = A and � � � � p � i=1 f(xi)|Ci| − � ∂A v · νA dHn−1 � � � � < ε . Proof. By Remark 2.3.3, the flux F : B �→ � ∂B v · νB dHn−1 is defined on the family of all figures B ⊂ A. In view of Remark 2.3.2, there are numbers 0 ≤ sk < 1 and disjoint, possibly empty, sets Ek ⊂ A such that v is pointwise Lipschitz in A − �∞ i=1 Ek, and for k = 1, 2, . . . , the following conditions hold: (i*) Hn−1+sk (Ek) < ∞, and Hsk v(x) < ∞ for each x ∈ Ek; (ii) Hn−1+sk (Ek) > 0 implies Hsk v(x) = 0 for each x ∈ Ek. © 2012 by Taylor & Francis Group, LLC
- 46. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 29 — #39 ✐ ✐ ✐ ✐ ✐ ✐ 2.3. Admissible maps 29 By Stepanoff’s theorem, A − �∞ i=1 Ek is the union of disjoint sets E0 and D ⊂ int A such that Hn (E0) = 0 and v is differentiable at each x ∈ D. Thus A is the union of disjoint sets D, E0, E1, . . . , and we let s0 = 1. The family � (Ek, sk) : k = 0, 1, . . . � is the disjoint union of subfamilies � (Ek, sk) : Hn−1+sk (Ek) > 0 � and � (Ek, sk) : Hn−1+sk (Ek) = 0 � , which we enumerate as � (E+ i , s+ i ) : i ≥ 1 � and � (E0 i , s0 i ) : i ≥ 1 � , respectively. For i, j ∈ N, let E0 i,j := � x ∈ E0 i : j − 1 ≤ Hs0 i v(x) < j � and define t+ i := n − 1 + s+ i and t0 i := n − 1 + s0 i . Now A − D is the union of disjoint sets E+ i and E0 i,j. Select ci > Ht+ i (E+ i ) and choose ε > 0. By Corollary 2.1.3, there is γ : A → R+ such that for each cube C ⊂ A, the following conditions are satisfied: (1) � �f(x)|C| − F(C) � � ≤ ε|C| if d(C) < γ(x) for some x ∈ D ∩ C, (2) � �F(C) � � ≤ ε2−i c−1 i d(C)t+ i if d(C) < γ(x) for some x ∈ E+ i ∩ C, (3) � �F(C) � � ≤ 2nj d(C)t0 i if d(C) < γ(x) for some x ∈ E0 i,j ∩ C. Next choose δ : A → R+. With no loss of generality, we may assume that δ ≤ γ. According to Proposition 2.2.2, there is a δ-fine dyadic partition P := � (C1, x1), . . . , (Cp, xp) � such that [P] = A and for κ = κ(n) > 0, � xk∈E+ i d(Ck)t+ i ≤ κci and � xk∈E0 i,j d(Ck)t0 i ≤ εj−1 2−i−j . Since f(x) = 0 for each x ∈ A − D, these inequalities, conditions (1)–(3), and Observation 2.1.1 imply the lemma: � � � � p � k=1 f(xk)|Ck| − � ∂A v · νA dHn−1 � � � � ≤ � xk∈D � � �f(xk)|Ck| − F(Ck) � � � + � xk∈A−D � �F(Ck) � � ≤ ε � xk∈D |Ck| + � i≥1 � � xk∈E+ i � �F(Ck) � � + ∞ � j=1 � xk∈E0 i,j � �F(Ck) � � � ≤ ε|A| + ε � i≥1 � 2−i c−1 i � xk∈E+ i d(Ck)t+ i + 2n ∞ � j=1 j � xk∈E0 i,j d(Ck)t0 i � ≤ ε|A| + εκ ∞ � i=1 2−i + 2nε ∞ � i,j=1 2−i−j = ε � |A| + κ + 2n � . © 2012 by Taylor & Francis Group, LLC
- 47. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 30 — #40 ✐ ✐ ✐ ✐ ✐ ✐ 30 2. Divergence theorem for dyadic figures Theorem 2.3.7. Let A be a dyadic figure. If v ∈ Adm(A; Rn ) is such that div v belongs to L1 (A), then � A div v dLn = � ∂A v · νA dHn−1 . Proof. Defining f as in Lemma 2.3.6, we have f ∈ L1 (A) and � A f dLn = � A div v dLn . Choose ε > 0, and select a function δ : A → R+ associated with ε and f according to Henstock’s lemma. By Lemma 2.3.6, there is a δ-fine partition P := � (C1, x1), . . . , (Cp, xp) � such that [P] = A and � � � � � A div v dLn − � ∂A v · νA dHn−1 � � � � ≤ � � � � � A f dLn − p � i=1 f(xi)|Ci| � � � � + � � � � p � i=1 f(x)|Ci| − � ∂A v · νA dHn−1 � � � � < 2ε. Remark 2.3.8. Some comments are in order. (1) The assumptions of Theorem 2.3.7 are met if v ∈ Lip(A; Rn ), since �div v�L∞(A) ≤ nLip v. (2) Let v(0) := 0, and v(x) := x cos |x|−n−1 for x ∈ Rn − {0}. Then v ∈ Adm(Rn ; Rn ), but div v does not belong to L1 (A) if A is a figure containing 0. Still, the flux of v can be calculated from div v by an averaging process which extends the Lebesgue integral. For a deeper analysis of this phenomenon, we refer the interested reader to [49, 51]; also see Chapter 9 below. (3) Assume n = 1, and let v : R → R be differentiable almost everywhere and such that � b a v� dL1 = v(b) − v(a) for each dyadic cell [a, b] ⊂ R. Since [29, Section 2.4.3] implies Hs �� x ∈ R : Hsv(x) > 0 �� = 0 for each 0 ≤ s < 1, condition (ii) of Definition 2.3.1 cannot be omit- ted. The Cantor-Vitali function (Example 9.2.5 below) and its multi- dimensional analogue [54] provide another rationale for the definition of admissible vector fields. (4) It is clear that using essentially the same arguments, the divergence theorem can be established for arbitrary figures. We employed dyadic figures merely for convenience. Theorem 2.3.7 is only a preliminary © 2012 by Taylor & Francis Group, LLC
- 48. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 31 — #41 ✐ ✐ ✐ ✐ ✐ ✐ 2.4. Convergence of dyadic figures 31 result, sufficient for proving a satisfactory integration by parts theorem stated below. Theorem 2.3.9 (Integration by parts). Let Ω ⊂ Rn be an open set, and let both v : Ω → Rn and g : Ω → R be locally bounded and pointwise Lipschitz almost everywhere in Ω. Assume div v ∈ L1 loc(U) and Dg ∈ L1 loc(Ω; Rn ). If gv ∈ Adm(Ω; Rn ) and spt (gv) � Ω, then � Ω g(x) div v(x) dx = − � Ω Dg(x) · v(x) dx. Proof. Let w = gv, and let A ⊂ Ω be a dyadic figure with spt w ⊂ int A. By the assumptions, v and g are measurable and bounded in A. Thus div w = g div v + Dg · v belongs to L1 (A). Since w � ∂A = 0, Theorem 2.3.7 yields 0 = � A div w(x) dx = � A g(x) div v(x) dx + � A Dg(x) · v(x) dx. (∗) Let x ∈ Ω − A. Then div w(x) = 0, since w vanishes in the open setΩ − A. Thus g(x) div v(x) = −Dg(x) · v(x) and either g(x) = 0 or v(x) = 0. This shows that spt (g div v) and spt (Dg · v) are subsets of A, and the theorem follows from (∗). Remark 2.3.10. The integration by parts theorem is usually applied when both g and v are admissible, and either g or v has compact support contained in Ω; see Proposition 2.3.5. 2.4. Convergence of dyadic figures As figures are too specialized for applications, it is desirable to extend the divergence theorem to a larger family of sets. With the sole purpose of enhancing intuition, we describe the first step of the most obvious approach to this problem. Our main results will be obtained from less obvious but more efficient ideas of R. Caccioppoli [14] and E. De Giorgi [18, 19]. Lemma 2.4.1. Let {Ai} be a sequence of measurable set, and let E ⊂ Rn be any set. If lim |E � Ai| = 0, then E is measurable and for each f ∈ L1(Rn), lim � Ai f(x) dx = � E f(x) dx. Proof. Since lim |E − Ai| = lim |Ai − E| = 0, passing to a subsequence if necessary, we may assume that |E − Ai| ≤ 2−i and |Ai − E| ≤ 2−i for i = 1, 2, . . . . Letting I := lim inf Ai = ∞ � j=1 ∞ � i=j Aj and S := lim sup Ai = ∞ � j=1 ∞ � i=j Aj, we infer |E − I| = |S − E| = 0. As I ⊂ S implies |E � I| = |E � S| = 0, the set E is measurable and lim �χE − χAi �L1(Rn) = 0. Any subsequence {Bi} of {Ai} has a © 2012 by Taylor & Francis Group, LLC
- 49. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 32 — #42 ✐ ✐ ✐ ✐ ✐ ✐ 32 2. Divergence theorem for dyadic figures subsequence {Ci} such that lim χCi = χE almost everywhere. Thus for f ∈ L1(Rn), the dominated convergence theorem yields lim � � � � � E f(x) dx − � Ci f(x) dx � � � � ≤ lim � Rn � �χE(x) − χCi (x) � � · � �f(x) � � dx = 0. The lemma follows from the arbitrariness of {Bi}. Recall that the family of all dyadic figures in Rn is denoted by DF. We say that a sequence {Ai} in DF converges to a set E ⊂ Rn if the following conditions are satisfied: (i) Each Ai is contained in a fixed compact set K ⊂ Rn. (ii) lim |Ai � E| = 0 and sup Hn−1(∂Ai) < ∞. By Lemma 2.4.1, the set E is measurable. Given dyadic figures A and B, we define nonoverlapping dyadic figures A � B := cl (A − B) and A ⊙ B = cl � int (A ∩ B)], and observe that A = (A � B) ∪ (A ⊙ B). Proposition 2.4.2. Let F be the flux of v ∈ C(Rn; Rn), and let {Ai} be a sequence in DF converging to a set E ⊂ Rn. There exists a finite limit � F(E) := lim F(Ai), which does not depend on the choice of the sequence {Ai}. Proof. Let K ⊂ Rn be a compact set containing all figures Ai, and let c = sup Hn−1(∂Ai). Choose ε > 0, and use the Stone-Weierstrass theorem [62, Theorem 7.32] to find a vector field w ∈ C1(Rn; Rn) such that �v − w�L∞(K;Rn) ≤ ε. According to Theorem 2.3.7, � �F(Ai � Aj) � � ≤ � � � � � ∂(Ai�Aj ) (v − w) · νAi�Aj dHn−1 � � � � + � � � � � ∂(Ai�Aj ) w · νAi�Aj dHn−1 � � � � ≤ εHn−1 � ∂(Ai � Aj) � + � � � � � Ai�Aj div w(x) dx � � � � ≤ 2cε + �div w�L∞(K)|Ai � Aj|; since ∂(Ai � Aj) ⊂ ∂Ai ∪ ∂Aj. By Observation 2.1.1, � �F(Ai) − F(Aj) � � = � � � � F(Ai � Aj) + F(Ai ⊙ Aj) � − � F(Aj � Ai) + F(Aj ⊙ Ai) �� � � ≤ � �F(Ai � Aj) � � + � �F(Aj � Ai) � � ≤ �div w�L∞(K) � |Ai � Aj| + |Aj � Ai| � + 4cε = �div w�L∞(K)|Ai � Aj| + 4cε. It follows that � F(Ai) � is a Cauchy sequence. The value � F(E) := lim F(Ai) does not depend on the sequence {Ai} of dyadic figures converging to E. Indeed, if {Bi} is another sequence of dyadic figures converging to E, then so does the interlaced sequence {A1, B1, A2, B2, . . . }, and consequently lim F(Ai) = lim F(Bi). Denote by DF the family of all sets E ⊂ Rn, necessarily measurable, for which there is a sequence {Ai} in DF converging to E. Clearly DF ⊂ DF, and since © 2012 by Taylor & Francis Group, LLC
- 50. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 33 — #43 ✐ ✐ ✐ ✐ ✐ ✐ 2.4. Convergence of dyadic figures 33 ∂(A ∪ B) ∪ ∂(A � B) ∪ ∂(A ⊙ B) ⊂ ∂A ∪ ∂B, it is easy to verify that the family DF is closed with respect to unions, intersections, and set differences. It follows from Proposition 2.4.2 that the flux F of v ∈ C(Rn; Rn) defined on figures in Section 2.1 has a unique extension � F : DF → R, still called the flux of v. Proposition 2.4.3. If � F : DF → R is the flux of v ∈ C(Rn; Rn), then � F(A ∪ B) = � F(A) + � F(B) for each pair of nonoverlapping sets A, B ∈ DF. Proof. Let {Ak} and {Bk} be sequences in DF that converge to A and B, respectively. From (Ak − Bk) − A ⊂ Ak − A and A − (Ak − Bk) = (A − Ak) ∪ (A ∩ Bk) ⊂ (A − Ak) ∪ (A ∩ B) ∪ (Bk − B), we infer lim � �A � (Ak � Bk) � � = 0. As ∂(Ak � Bk) ⊂ ∂Ak ∪ ∂Bk, the sequences {Ak � Bk} and � (Ak � Bk) ∪ Bk � = {Ak ∪ Bk} converge to A and A ∪ B, respectively. Thus � F(A ∪ B) = lim F � (Ak � Bk) ∪ Bk � = lim F(Ak � Bk) + lim F(Bk) = � F(A) + � F(B). Proposition 2.4.4. Let � F : DF → R be the flux of v ∈ C(Rn; Rn). If v is admissible and div v belongs to L1 loc(Rn), then for each E ∈ DF, � F(E) = � E div v(x) dx. Proof. If {Ai} is a sequence of dyadic figures converging to E, then Proposition 2.4.2, Theorem 2.3.7, and Lemma 2.4.1 imply � F(E) = lim F(Ai) = lim � Ai div v(x) dx = � E div v(x) dx. Proposition 2.4.4 establishes the divergence theorem for sets in DF, which are the desired generalization of dyadic figures (cf. Corollary 6.7.4 below). However, the flux � F : DF → R does not share the geometric content of the flux F : DF → R defined by formula (2.1.1). This will be remedied in Chapters 4–6 below, albeit with a substantial effort. We show that each set E ∈ DF has an “essential boundary” ∂∗E ⊂ ∂E and a “unit exterior normal” νE, defined Hn−1 almost everywhere on ∂∗E, such that the flux � F of a vector field v ∈ C(Rn; Rn) is calculated by the formula � F(E) = � ∂∗E v · νE dHn−1 analogous to (2.1.1); see formula (6.5.1) below. Remark 2.4.5. Using the convergence of dyadic figures, it is possible to define a sequential topology T in DF that is induced by a uniformity, and show that DF is the sequential completion of the space (DF, T); see Chapter 10, in particular Section 10.6. Since the flux F : (DF, T) → R of v ∈ C(Rn; Rn) is uniformly continuous by additivity, it has a unique continuous extension � F : DF → R — a fact we proved directly in Proposition 2.4.2. © 2012 by Taylor & Francis Group, LLC
- 52. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 35 — #45 ✐ ✐ ✐ ✐ ✐ ✐ Chapter 3 Removable singularities We will study removable singularities for the Cauchy-Riemann, Laplace, and minimal surface equations. As these equations are in the divergence form div � φ(Du) � = 0, the integration by parts theorem established in the previous chapter is a natural tool. We define removable sets by means of Hausdorff measures, mostly without any topological restrictions. The results are estab- lished by short and simple arguments, which rely on the relationship between weak and strong solutions of partial differential equations. A few basic facts about distributions and weak solutions are stated without proofs. We made no attempt to survey the long history concerning removable singularities. 3.1. Distributions A multi-index is an n-tuple α := (α1, . . . , αn) where αi are nonnegative inte- gers. Let |α| := �n i=1 αi and Dα := Dα1 1 · · · Dαn n = � ∂ ∂ξ1 �α1 · · · � ∂ ∂ξn �αn . Note that if |α| = 0, then Dα f = f for any f : Rn → C. LetΩ ⊂ Rn be an open set. Employing convention (1.1.2), we say that a sequence {ϕi} in C∞ c (Ω; C) converges to zero in the sense of test functions if the following conditions hold: (i) {ϕi} is a sequence in C∞ c (U; C) for an open set U � Ω; (ii) lim �Dα ϕi�L∞(Ω;C) = 0 for each multi-index α. The complex linear space C∞ c (Ω; C) equipped with this convergence is de- noted by D(Ω; C), and the elements of D(Ω; C) are called test functions. The real linear subspace of D(Ω; C) consisting of all real-valued test functions is denoted by D(Ω). A distribution is a linear functional L : D(Ω; C) → C such that lim L(ϕi) = 0 35 © 2012 by Taylor & Francis Group, LLC
- 53. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 36 — #46 ✐ ✐ ✐ ✐ ✐ ✐ 36 3. Removable singularities for each sequence {ϕi} in D(Ω; C) that converges to zero in the sense of test functions. The complex linear space of all distributions is denoted by D� (Ω; C). The real linear space D� (Ω) is defined analogously. Remark 3.1.1. In Example 3.6.5 below we define a locally convex topology S in the spaces D(Ω) of real-valued test functions so that the space D�(Ω) of distributions is the dual space of � D(Ω), S). The reader familiar with complex locally convex spaces will recognize instantly that a similar topology can be defined in the space D(Ω; C) of complex-valued test functions [64, Section 6.2]. Example 3.1.2. Let f ∈ L1 loc(Ω; C), let µ be a Radon measure in Ω, and let v ∈ L1 loc(Ω; Rn ). The linear functionals Lf : ϕ �→ � Ω fϕ dx : D(Ω; C) → C, (1) Lµ : ϕ �→ � Ω ϕ dµ : D(Ω; C) → C, (2) Fv : ϕ �→ − � Ω v · Dϕ dx : D(Ω) → R (3) are examples of distributions. Distribution Fv is called the distributional divergence of v, since if v ∈ C1 (Ω; Rn ) integration by parts yields Fv(ϕ) = � Ω ϕ div v dx = Ldiv v(ϕ) for each test function ϕ ∈ D(Ω). Let α be a multi-index. If f ∈ C|α| (Ω; C), then repeated applications of the integration by parts theorem yield �LDαf , ϕ� = � Ω ϕ(x)Dα f(x) dx = (−1)|α| � Ω f(x)Dα ϕ(x) dx = (−1)|α| �Lf , Dα ϕ� for each ϕ ∈ D(Ω; C). Since for any distribution L, the linear functional ϕ �→ (−1)|α| �L, Dα ϕ� : D(Ω; C) → C is a distribution, the previous identity suggests to define a distribution Dα L by the formula �Dα L,ϕ � := (−1)|α| �L, Dα ϕ� for each test function ϕ ∈ D(Ω; C). Observe that Dα Lf = LDαf whenever f ∈ C|α| (Ω; C). Additional information about test functions and distributions can be found in many standard textbooks, for instance in [64, Chapter 6] or [27, Chapter 5]. © 2012 by Taylor & Francis Group, LLC
- 54. ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 37 — #47 ✐ ✐ ✐ ✐ ✐ ✐ 3.2. Differential equations 37 3.2. Differential equations A linear differential operator with constant coefficients is the expression Λ := � |α|≤k cαDα where k ∈ N and cα ∈ C for each multi-index α with |α| ≤ k. In an open set Ω ⊂ Rn we consider two types of solutions of the partial differential equation Λu = 0. • A strong solution is a complex-valued function u ∈ Ck (Ω; C) such that � Λu, x � = 0 for each x ∈ Ω. • A weak solution is a complex-valued function u ∈ L1 (Ω; C) such that ΛLu = 0 where Lu is defined in Example 3.1.2, (1). Thus u ∈ L1 (Ω; C) is a weak solution of Λu = 0 if the equality � |α|≤k (−1)|α| cα � Ω u(x)Dα ϕ(x) dx = � ΛLu, ϕ � = 0 holds for each test function ϕ ∈ D(Ω; C). For a multi-index α = (α1, . . . , αn) and x = (ξ1, . . . , ξn) in Rn , let xα := ξα1 1 · · · ξαn n where ξαi = 1 when αi = 0. A complex-valued function pΛ : x �→ � |α|=k cαxα : Rn → C is called the characteristic polynomial of Λ. If pΛ(x) �= 0 for each x in Rn −{0}, the operator Λ is called elliptic. Note that the ellipticity of Λ is determined by the leading coefficients, i.e., by cα with |α| = k. Example 3.2.1. The following linear differential operators are elliptic. (1) The Laplace operator � := D2 1 + · · · + D2 n, since p�(x) = |x|2 for each x ∈ Rn . (2) For n = 2 and i := √ −1, the holomorphic operator ¯ ∂ := D1 + iD2, since p¯ ∂(x) = ξ1 + iξ2 for each x = (ξ1, ξ2) in R2 . © 2012 by Taylor & Francis Group, LLC
- 55. Another Random Scribd Document with Unrelated Content
- 56. relaxac'one & Pardonac'one p̶'fat' Joh̅ is Trenchard ac etiam p'litetr allocetr in om'ib: Curiis n'ris absq: aliquo Brevi de Allocac'one mea parte pr'm's obtent' sive obtinend'. Et non obstante aliqua def'tu vel aliquib' def'tibus in his l'ris n'ris patentib' content' aut aliquo statuto, acto, ordinac'one provisione seu Restricc'one aut aliqua al' re, causa, vel materia quacunq: in contrar' inde ullo modo non obstante. In Cuius rei testimoniu' has l'ras n'ras fier' fecimus Patentes. Teste me ip'o apud West' decimo sept'o die Decembris anno regni n'ri tertio. Per Breve de p'rato Sigillo Barker." This was in the year 1688, just seven days after, according to Macaulay, that he had fled secretly from the kingdom, having previously thrown the great seal into the Thames, whence it was dredged up some months after by a fisherman. Being driven back by stress of weather, he returned to London, and on the 17th Pepys states, "That night was a council; his Maty refuses to assent to all the proposals, goes away again to Rochester." and on that very night was this pardon granted, James probably endeavouring to prop up his tottering cause by attaching as many as possible to his own party. There were several documents in the collection of the late Josiah Trench, Esq., of Windsor (1648-1652) signed by John Trenchard, among the other regicides. Ewing, in his Norfolk Lists, states that a portrait of him is in existence, and that he was a serjeant-at-law, and at this date (1688) M. P. for Thetford, being at that date merely an esquire. In 1692, according to the same authority, Sir John Trenchard was Secretary of State; and his death took place in 1694. I should be glad to add to these scanty notices, especially as regards the reason which rendered a pardon necessary at this time. E. S. Taylor.
- 57. Replies to Minor Queries. Dayesman (Vol. i., p. 189.). —Bishop Jewell writes: "M. Harding would have had us put God's word to daying (i.e. to trial), and none otherwise to be obedient to Christ's commandment, than if a few bishops gathered at Trident shall allow it."—Replie to Harding, Works, vol. ii. p. 424. (Dr. Jelf's edit.) "The Ger. Tagen, to appoint a day. The D. Daghen, to cite or summon on a day appointed."— (Wachter and Kilian.) And Dayesman is he, the man, "who fixes the day, who is present, or sits as judge, arbiter, or umpire on the day fixed or appointed." It is evident that Richardson made much use of Jewell; but this word "daying" has escaped him: his explanation of dayesman accords well with it. Q. Bull; Dun (Vol. ii., p. 143.). —We certainly do not want the aid of Obadiah Bull and Joe Dun to account for these words. Milton writes, "I affirm it to be a bull, taking away the essence of that, which it calls itself." And a bull is, "that which expresses something in opposition to what is intended, wished, or felt;" and so named "from the contrast of humble profession with despotic commands of Papal bulls." "A dun is one who has dinned another for money or anything."— See Tooke, vol. ii. p. 305. Q.
- 58. Algernon Sidney (Vol. v., p. 447.). —I do not intend to enter the lists in defence of this "illustrious patriot." The pages of "N. & Q." are not a fit battle ground. But I request you to insert the whole quotation, that your readers may judge with what amount of fairness C. has made his note from Macaulay's History. "Communications were opened between Barillon, the ambassador of Lewis, and those English politicians who had always professed, and who indeed sincerely felt, the greatest dread and dislike of the French ascendancy. The most upright member of the country party, William Lord Russell, son of the Earl of Bedford, did not scruple to concert with a foreign mission schemes for embarrassing his own sovereign. This was the whole extent of Russell's offence. His principles and his fortune alike raised him above all temptations of a sordid kind: but there is too much reason to believe that some of his associates were less scrupulous. It would be unjust to impute to them the extreme wickedness of taking bribes to injure their country. On the contrary, they meant to serve her: but it is impossible to deny that they were mean and indelicate enough to let a foreign prince pay them for serving her. Among those who cannot be acquitted of this degrading charge was one man who is popularly considered as the personification of public spirit, and who, in spite of some great moral and intellectual faults, has a just claim to be called a hero, a philosopher, and a patriot. It is impossible to see without pain such a name in the list of the pensioners of France. Yet it is some consolation to reflect that in our own time a public man would be thought lost to all sense of duty and shame who should not spurn from him a temptation which conquered the virtue and the pride of Algernon Sidney." History of England, vol. i. p. 228. Algernon Holt White.
- 59. Brighton. Age of Trees (Vol. iv., pp. 401. 488.). —At Neustadt, in Wirtemberg, there is a prodigious lime-tree, which gives its name to the town, which is called Neustadt an der Linden. The age of this tree is said to be 1000 years. According to a German writer, it required the support of sixty pillars in the year 1392, and attained its present size in 1541. It now rests, says the same authority, on above one hundred props, and spreads out so far that a market can be held under its shade. It is of this tree that Evelyn says it was— "Set about with divers columns and monuments of stone (eighty-two in number, and formerly above one hundred more), which several princes and nobles have adorned, and which as so many pillars serve likewise to support the umbrageous and venerable boughs; and that even the tree had been much ampler the ruins and distances of the columns declare, which the rude soldiers have greatly impaired." There is another colossal specimen of the same species in the churchyard of the village of Cadiz, near Dresden. The circumference of the trunk is forty feet. Singularly, though it is completely hollow through age, its inner surface is coated with a fresh and healthy bark. Unicorn. Emaciated Monumental Effigies (Vol. v., p. 427.). —In reference to your correspondents' observations on skeleton monuments, I may mention that there is one inserted in the wall of the yard of St. Peter's Church, Drogheda. It is in high relief, cut in a dark stone and the skeleton figure half shrouded by grave clothes is a sufficiently appalling object. Beside it stands another figure still "in
- 60. the flesh." It is many years since I saw the monument, and whether there be any inscription legible upon it, or whether it be generally known to whom it belongs, I cannot inform you. Ursula. There is a very good instance of an "altar tomb," bearing on it an ordinary effigy, and containing within it a skeleton figure, visible through pierced panel work, in Fyfield Church, Berks. It is the monument of Sir John Golafre, temp. Hen. V. Another fine instance I remember to have seen (I believe) in the parish church of Ewelme, Oxon. Henry G. Tomkins. Weston-super-Mare. Bee Park (Vol. v., p. 322.). —In this neighbourhood is an ancient farm-house called Bee Hall, where I doubt not that bees were kept in great quantities in bygone ages; and am the more led to believe this because they always flourish best upon thyme, which grows here as freely and luxuriantly as I ever elsewhere observed it. About four miles from said Bee Hall, the other day, I was looking over a genteel residence, and noticing a shady enclosure, asked the gardener what it was for. He told me, to protect the bees from the sun: it was upon a much larger scale than we generally now see, indicating that the soil, &c. suit apiaries. Looking to the frequent mention of honey, and its vast consumption formerly, as you instance in royal inventories, to which may be added documents in cathedral archives, &c., is it not remarkable that we should witness so few memorials of the ancient management of this interesting insect? I certainly remember one well-built "bee- house," at the edge of Lord Portsmouth's park, Hurstbourne, Hants, large enough for a good cottage, now deserted. While on the subject I will solicit information on a custom well known to those resident in the country, viz. of making a great noise with a house key, or other small knocker, against a metal dish or kettle while bees are swarming? Of course farmers' wives, peasants, &c., who do not
- 61. reason, adopt this because their fathers before them did so. It is urged by intelligent naturalists that it is utterly useless, as bees have no sense of hearing. What does the clamour mean,—whence derived? B. B. Pembroke. Sally Lunn (Vol. v., p. 371.). —In reply to the Query, "Is anything known of Sally Lunn? is she a personage or a myth?" I refer your inquirer to Hone's Every-day Book, vol. ii. p. 1561.: "The bun so fashionable, called the Sally Lunn, originated with a young woman of that name at Bath, about thirty years ago." [This was written in 1826.] "She first cried them in a basket, with a white cloth over it, morning and evening. Dalmer, a respectable baker and musician, noticed her, bought her business, and made a song and set it to music in behalf of Sally Lunn. This composition became the street favourite, barrows were made to distribute the nice cakes, Dalmer profited thereby and retired, and to this day the Sally Lunn Cake claims pre-eminence in all the cities of England." J. R. W. Bristol. Baxter's Pulpit (Vol. v., p. 363.). —An engraving of Baxter's pulpit will be found in a work entitled Footsteps of our Forefathers: what they suffered and what they sought. By James G. Miall, 1851, p. 232. J. R. W. Bristol. Lothian's Scottish Historical Maps (Vol. v., p. 371.).
- 62. —Although this work is now out of print, and thereby scarce, your correspondent Elginensis will, I have no doubt, on application to Stevenson, the "well-known" antiquarian and historical bookseller in Edinburgh, be put in possession of a copy for 12s. T. G. P. Edinburgh. British Ambassadors (Vol. iv., pp. 442. 477.). —Some time ago a correspondent asked where he could obtain a list or lists of the ambassadors sent from this court. I do not recollect that an answer has appeared in your columns, nor do I know how far the following may suit his purpose: "12. An Alphabetical Index of the Names and Dates of Employment of English Ambassadors and Diplomatic Agents resident in Foreign Courts, from the Reign of King Henry VIII. to that of Queen Anne inclusive. One volume, folio." This is extracted from the letter of the Right Hon. H. Hobhouse, keeper of His Majesty's State Papers, in reply to the Secretary of the Commissioners of Public Records, dated "State Paper Office, Sept. 19, 1832." (See the Appendix to the Commissioners' Report, 1837, p. 78.) Tee Bee. Knollys Family (Vol. v., p. 397.). —Lt.-General William Knollys, eighth Earl of Banbury, married Charlotte Martha, second daughter of the Ebenezer Blackwell, Esq., banker, of Lombard Street, and Lewisham, Kent. The present Col. Knollys, of the Fusileer Guards, is his representative. A. Blackwell, sister or daughter of John Blackwell, the father of Ebenezer, married an Etheridge.
- 63. W. Blackwell, Curate of Mells. 'Prentice Pillars—'Prentice Windows (Vol. v., p. 395.). —I am reminded of a similar story connected with the two rose windows in the transept of the beautiful cathedral of Rouen. They were described to me by the old Swiss in charge, as the work of two artists, master and pupil; and he also pointed out the spot where the master killed the pupil, from jealousy of the splendid production of the north window by the latter: and, as the Guide Book truly says, "La rose du nord est plus belle que celle du midi"—the master's work. Benbow. Birmingham. St. Bartholomew (Vol. v., p. 129.). —Thanking you for the information given, may I further inquire if any of your correspondents are aware of the existence of any copy or print from the picture in the Church of Notre Dame, at Paris, of St. Bartholomew healing the Princess of Armenia (see Jameson's Sacred and Legendary Art); and where such may be seen? Regedonum. Sun-dial Inscription (Vol. v., p. 79.). —The following inscription is painted in huge letters over the sun- dial in front of an old farm-house near Farnworth in Lancashire: "Horas non numero nisi serenas." Where are these words to be found? Y. History of Faction (Vol. v., p. 225.).
- 64. —In my copy of this work, published in 1705, 8vo., formerly Isaac Reed's, he attributes it to Colonel Sackville Tufton. I observe also that Wilson (Life of De Foe, vol. ii. p. 335.) states, that in his copy it is ascribed, in an old handwriting, to the same author. Jas. Crossley. Barnacles (Vol. v., p. 13.). —May not the use of this word in the sense of spectacles be a corruption of binoculis; and has not binnacle (part of a ship) a similar origin? J. S. Warden. Family Likenesses (Vol. v., p. 7.). —Any one who mixed in the society of the Scottish metropolis a few years ago must have met with two very handsome and accomplished brothers, who generally wore the Highland dress, and were known by the name of "The Princes." I do not mean to enter into the question as to whether or not they were the true representatives of "Bonnie Prince Charlie," which most persons consider to have been conclusively settled in the negative by an article which appeared in the Quarterly Review: but most assuredly a very strong point of evidence in favour of their having the royal blood of Scotland in their veins, was the remarkable resemblance which they bore—especially the younger brother—to various portraits of the Stuart family, and, among the rest, to those of the "Merry Monarch," as well as of his father Charles I. E. N. Merchant Adventurers to Spain (Vol. v., p. 276.). —C.J.P. may possibly be assisted in his inquiries by referring to De Castros' Jews in Spain, translated by Kirwan, pp. 190-196. This
- 65. interesting work was published by G. Bell, 186. Fleet Street, London, 1851. W. W. La Valetta, Malta. Exeter Controversy (Vol. v., p. 126.). —This controversy was one of the many discussions relating to the Trinity which have engaged the theological activity of England during the last two hundred years. It arose in consequence of the imputed Arianism of some Presbyterian ministers of Exeter, the most conspicuous of whom were James Peirce and Joseph Hallet. It began in 1717, and terminated in 1719, when these two ministers were ejected from their pulpits. Your correspondent who put the question will find some account of this controversy in Murch's History of the Presbyterian Churches in the West of England,—a work well worth the attention of those who take interest in the antiquities of Non- conformity. T. H. Gill. Corrupted Names of Places (Vol. v., p. 375.). —When my father was at one time engaged in collecting the numbers drawn for the Sussex militia, he began by calling out for those men who belonged to the hundred of Mayfield; and though he three times repeated his call, not a single man came forward. A person standing by suggested that he should say "the hundred of Mearvel," and give it as broad a twang as possible. He did so; when nineteen out of twenty-three present answered to the summons. Hurstmonceaux is commonly pronounced Harsmouncy; and I have heard Sompting called Summut. G. Blink. Poison (Vol. v., p. 394.).
- 66. —Junius, Bailey, and Johnson seem all to agree that our word poison comes from the French poison. I am inclined to think, with the two first-mentioned lexicographers, that the etymon is πόσις, or potio. Junius adds, that "Ita Belgis venenum dicitur gift, donum;" and it is curious that in Icelandic eitr means both poison and gift. In the Antiquitates Celto-Scandicæ (p. 13.), I find the following expressions:—"Sva er sagt, at Froda væri gefinn banadryckr." "Mixta portioni veneno sublatum e vivis tradunt Frotonem." Should it not be potioni, inasmuch as "bana," in Icelandic, signifies to kill, if I do not err, and "dryckr" is drink? Certainly, in Anglo-Saxon, "bana" (whence our bane) and "drycian" have similar significations. C. I. R. Is there any possible doubt that poison is potion? Menage quotes Suetonius, that Caligula was potionatus by his wife. It is a French word undoubtedly. C. B. Vikingr Skotar (Vol. v., p. 394.). —In the Antiquitates Celto-Scandicæ it is stated (p. 5.), that after the death of Guthormr, and subsequently to the departure of Harald (Harfagr) from the Hebrides, "Sidan settug i löndin vikingar margir Danir oc Nordmenn. Posthac sedes ibi occupant piratæ plurimi, Dani æqua ac Normanni." The word vikingar, the true Icelandic word for pirate, often occurs in the same saga, but not combined with skotar, though this latter term is repeated, signifying "the Scotch," and also in composition with konungr, &c. C. I. R. Rhymes on Places (Vol. v., pp. 293. 374.). —A complete collection of local rhymes would certainly be both curious and interesting. Those cited by Chambers in his amusing work are exclusively Scotch; for a collection relating to English towns, I would refer your Querist Mr. Fraser to Grose's Provincial
- 67. Glossary, where, interspersed among the "Local Proverbs," he will find an extensive gathering of characteristic rhymes. I conclude with appending a few not to be found in either of these works: "RICHMOND. "Nomen habes mundi, nec erit sine jure, secundi, Namque situs titulum comprobat ipse tuum. From thy rich mound thy appellation came, And thy rich seat proves it a proper name." Drunken Barnaby's Journal. "Anglia, mons, fons, pons, ecclesia, fœmina, lana. England amongst all nations is most full, Of hills, wells, bridges, churches, women, wool." Ibid. "Cornwall swab-pie, and Devon white-pot brings, And Leicester beans, and bacon fit for kings." Dr. King's Art of Cookery. See Spectator. In Belgium I am perhaps beyond bounds, but may cite in conclusion: "Nobilibus Bruxella viris, Antverpia nummis, Gandavum laqueis, formosis Burga puellis, Lovanium doctis, gaudet Mechlinia stultis." William Bates. You may perhaps think the accompanying, "Rhymes on Places" worthy of insertion, on the districts of the county of Ayr, viz.: "Carrick for a man, Kyle for a cou, Cunninghame for butter and cheese, And Galloway for woo." F. J. H.
- 68. "We three" (Vol. v., p. 338.). —It may interest your correspondent to learn that a public-house exists in London with the sign he mentions. It is situate in Virginia Row, Bethnal Green, is styled "The Three Loggerheads," and has a signboard ornamented with a couple of busts: one of somewhat Cæsarian aspect, laureated; the other a formidable-looking personage with something on his head, probably intended for the dog-skin helmet of the ancient Greeks,—but as the style of art strongly reminds one of that adopted for the figure-heads of ships, I confess my doubts on the subject. Under each bust appears the distich: "WE THREE LOGGERHEADS BE." The sign appears a "notability" in the neighbourhood, as I have more than once in passing seen some apparent new comer set to guess its meaning; and when he confessed his inability, informed, in language more forcible than elegant, that he made the third Loggerhead. W. E. F. Burning Fern brings Rain (Vol. v., p. 242.). —In some parts of America, but more particularly in the New England States, there was a popular belief, in former times, that immediately after a large fire in a town, or of wood in a forest, there would be a "fall of rain." Whether this opinion exists among the people at present, or whether it was entertained by John Winthrop, the first governor of the colony of Massachusetts Bay, and the Pilgrim Fathers, on their landing at Plymouth, as they most unfortunately did, their superstitious belief in witchcraft, and some other "strange notions," may be a subject of future inquiry. W. W. La Valetta, Malta.
- 69. Plague Stones (Vol. v., pp. 226. 374.). —I have often seen the stone which G. J. R. G. mentions as "to be seen close to Gresford, in Denbighshire, about a quarter of a mile from the town, on the road to Wrexham, under a wide-spreading tree, on an open space, where three roads meet." It is, I conjecture, the base of a cross. This stone may be the remnant of the last of a succession of crosses, the first of which may have given its Welsh name, Croes ffordd, the way of the cross, to the village. There is no tradition of any visitation of the plague at Gresford; but there is reason to suppose that it once prevailed at Wrexham, which is about three miles distant. Near that town, and on the side of a hill near the footpath leading from Wrexham vechan to Marchwiel Hall, there is a field called Bryn y cabanau, the brow of the cabins; the tradition respecting which is, that, during the prevalence of the plague in Wrexham, the inhabitants constructed wooden huts in this place for their temporary residences. A Quondam Gresfordite. I do not think the "Plague Stone" a mile or two out of Hereford has been mentioned in the Notes on that subject. If my memory is correct, there is a good deal of ornament, and it is surrounded by a short flight of stone steps. F. J. H. Sneezing (Vol. v., p. 364.). —Having occasion to look at the first edition of the Golden Legend, printed by Caxton, I met with the following passage, which may perhaps prove interesting to your correspondent, as showing that the custom of blessing persons when they sneeze "endured" in the fifteenth century. The institution of the "Litany the more and the lasse," we are told, was justified,— "For a right grete and grevous maladye: for as the Romayns had in the lenton lyued sobrely and in contynence, and after
- 70. at Ester had receyud theyr Sauyour; after they disordered them in etyng, in drynkyng, in playes, and in lecherye. And therfore our Lord was meuyed ayenst them and sente them a grete pestelence, which was called the Botche of impedymye, and that was cruell and sodayne, and caused peple to dye in goyng by the waye, in pleying, in leeyng atte table, and in spekyng one with another sodeynly they deyed. In this manere somtyme snesyng they deyed; so that whan any persone was herd snesyng, anone they that were by said to hym, God helpe you, or Cryst helpe, and yet endureth the custome. And also when he sneseth or gapeth he maketh to fore his face the signe of the crosse and blessith hym. And yet endureth this custome." Golden Legende, edit. 1483, fo. xxi. b. F. Somner Merryweather. Kentish Town. Abbot of Croyland's Motto (Vol. v., p. 395.). —Mr. Forbes is quite correct with regard to the motto of Abbot Wells, which should be "Benedicite Fontes Domino." The sentence, "Bless the Wells, O Lord!" which is placed in so awkward a juxtaposition with it, is really a distinct motto for the name of Wells, and, so far from being a translation of the abbot's, is almost an inversion of it; and this should, as Mr. Forbes justly remarks, have had "some editorial notice" from me. M. A. Lower. Derivation of the Word "Azores" (Vol. v., p. 439.). —The group of islands called the Azores, first discovered in 1439, by Joshua Vanderburg, a merchant of Bruges, and taken possession of by the Portuguese in 1448, were so named by Martin Behem, from the Portuguese word Açor, a hawk; Behem observing a great number of hawks there. The three species most frequently seen now
- 71. are the Kestril, called Francelho; the Sparrowhawk, Furobardo; and the Buzzard, Manta; but whether very numerous or not, I am unable to state. From the geographical position of these islands, correct lists of the birds and fishes would be of great interest, and, as far as I am aware, are yet wanting. Martin Behem found one of these islands covered with beech- trees, and called it therefore Fayal, from the Portuguese word Faya, a beech-tree. Another island, abounding in sweet flowers, he called Flores, from the Portuguese, Flor, a flower. Terceira, one of the nine islands forming the group, is said to have been so called, because, in the order of succession, it was the third island discovered (from Ter and ceira, a bank). Graciosa, as a name, was conferred upon one of peculiar beauty, a sort of paradise. Pico derived its name from its sugar-loaf form. The raven found at Madeira and the Canary Islands is probably also a native of the Azores, and might have suggested the Portuguese name of Corvo for one of the nine. St. Mary, St. Michael, and St. George complete the names of the group, of which St. Michael is the largest and Corvo the smallest. Wm. Yarrell. Rider Street. Scologlandis and Scologi (Vol. v., p. 416.). —As these names occur in a Celtic country, we are justified in seeking their explanation in the Celtic language. I therefore write to inform G. J. R. G. that the word scolog is a living word in the Irish language, and that it signifies a farmer or husbandman. It is the word used in the Irish Bible at Matt. xxi. 33., "he let it out to husbandmen"—tug se do scologaibh ar chios i. I may also mention that the name Mac Scoloige is very common in the co. Fermanagh in Ireland, where it is very generally anglicised Farmer, according to a usual practice of the Irish. Thus it is not uncommon even now to find a man known by the name of John or Thomas Farmer, whose father or grandfather is John or Thomas Mac Scoloige, the name Mac Scoloige signifying "son of a farmer."
- 72. The Scologlandis, in the documents quoted by G. J. R. G., must therefore have taken their name from the scologs or farmers, by whom they were cultivated, unless we suppose that they were anciently the patrimony of some branch of the family of Mac Scoloige, whose remains are now settled in Fermanagh. In Scotland the word is now usually written sgalag, and is explained by Armstrong in his Gaelic Dictionary "a farm servant." And the word does certainly seem to have been used in ancient Irish to denote a servant or menial attendant, although the notion of a farm servant seems to have grown out of its other significations. Thus in a very ancient historical romance (probably as old as the ninth or tenth century), which is preserved in the curious volume called Leabhar breac, or Speckled Book, in the library of the Royal Irish Academy, the word scolog is used to designate the servant of the Abbot of St. Finbar's, Cork. J. H. T.
- 73. Miscellaneous. NOTES ON BOOKS, ETC. If there be any one class of documents from which, more than from any other, we may hope to draw evidence of the accuracy of Byron's assertion, that "Truth is strange, stranger than fiction!" they are surely the records of judicial proceedings both in civil and criminal matters; while, as Mr. Burton well observes in the preface to the two volumes which have called forth this remark, Narratives from Criminal Trials in Scotland, "there can be no source of information more fruitful in incidents which have the attraction of picturesqueness, along with the usefulness of truth." In submitting therefore to the public the materials of this nature—some drawn from manuscript authorities, some again from those works which, being printed for Subscription Clubs, may be considered as privately printed, and inaccessible to the majority of readers—which had accumulated on his hands while in the pursuit of other inquiries connected with the history of Scotland, Mr. Burton has produced two volumes which will be read with the deepest interest. The narratives are of the most varied character; and while some give us strange glimpses of the workings of the human heart, and show us how truly the Prophet spoke when he described it as being "deceitful above all things, and desperately wicked;" and some exhibit humiliating pictures of the fallibility of human judgment, others derive their chief interest from revealing collaterally "the social secrets of the day,— from the state mysteries, guarded by the etiquette and policy of courts, down to those characteristics of humble life which are removed from ordinary notice by their native obscurity." Greater
- 74. dramatic power on the part of Mr. Burton might have given additional attraction to his narratives; but though the want of this power is obvious, they form two volumes which will be perused with great curiosity and interest even by the most passionless of readers. Speaking of the use of Records reminds us that our valued cotemporary The Athenæum has anticipated us in a purpose we have long entertained, of calling the attention of historical inquirers to the vast amount of new material for illustrating English history to be found in Sir F. Palgrave's Calendar of the "Baga de Secretis," printed by him in several of his Reports, as Deputy Keeper of the Records. As The Athenæum has however entered upon the subject, we cannot do better than refer our readers to its columns. Letter addressed to Lord Viscount Mahon, M.P., President of the Society of Antiquaries, on the Propriety of Reconsidering the Resolutions of that Society which regulate the Payments from the Fellows: by John Bruce, Esq., Treas. S.A.—is the title of a temperate and well-argued endeavour on the part of the Treasurer, to persuade the Society of Antiquaries to return to that scale of subscription, &c. which prevailed at the moment when unquestionably the Society was at its highest point of reputation and usefulness. Originally addressed to the President, and then communicated to the Council, it has now been submitted to the Fellows, that they may see some of the grounds on which the Council have recommended, and on which they are invited to ballot on Thursday next, in favour of a reversal of the Resolution of 1807. Looking to the general state and prosperity of the Society as exhibited in this pamphlet, and comparing the payments to it with those to the numerous Archæological Societies which have sprung up of late years, the proposal seems to be well-timed, and deserving to be adopted by the Fellows as obviously calculated to extend the usefulness and raise the character of the Society. We hope that when the ballot is taken, some of those old friends of the Society to whose former exertions, in connexion with its financial arrangements, the Society owes so much, and who are understood now to be doubtful as to the measure, will put in their white balls in favour of a step which
- 75. ought clearly to lead to increased exertions on the part of all persons connected with the Society; and which may well be advocated on the ground, that it must lead to such a result. The lovers of elaborate and highly finished drawings of antiquarian objects are recommended to inspect some specimens of Mr. Shaw's artistic skill, comprising portraits of Mary Queen of Scots, Mary of England, the Pall of the Fishmongers' Company, which will be on view to-day and Monday at Sotheby and Wilkinson's Rooms, previous to their sale by auction on Tuesday next. BOOKS AND ODD VOLUMES WANTED TO PURCHASE. Biblia Sacra, Vulg. Edit., cum Commentar. Menochii. Alost and Ghent, 1826. Vol. I. Barante, Ducs de Bourgogne. Vols. I. and II. 1st, 2nd, or 3rd Edit. Paris Ladvocat, 1825. Biographia Americana, by a Gentleman of Philadelphia. Potgieseri de Conditione Servorum Apud Germanos. 8vo. Col. Agrip. The British Poets. Whittingham's edition in 100 Vols., with plates. Repository of Patents and Inventions. Vol. XLV. 2nd Series. 1824. —— Vol. V. 3rd Series 1827. Nicholson's Philosophical Journal. Vols. XIV. XV. 1806. Journal of the Royal Institution of Great Britain. No. XI. 2nd Series. Sorocold's Book of Devotions. Works of Isaac Barrow, D.D., late Master of Trinity College, Cambridge. London, 1683. Vol. I. Folio. Lingard's History of England. Vols. VI. VII. VIII. IX. XII. XIII., Cloth. Fabricii Bibliotheca Latina. Ed. Ernesti. Leipsig, 1773. Vol. III. The Anacalypsis. By Godfrey Higgins. 2 Vols. 4to.
- 76. Codex Diplomaticus Ævi Saxonici, opera J. M. Kemble. Vols. I. and II. 8vo. Eckhel, Doctrina Numorum. Vol. VIII. Brougham's Men of Letters. 2nd Series, royal 8vo., boards. Original edition. Knight's Pictorial Shakspeare. Royal 8vo. Parts XLII. XLIII. XLIV. L. and LI. Conder's Analytical View of all Religions. 8vo. Halliwell on the Dialects of Somersetshire. Sclopetaria, or Remarks on Rifles, &c. *⁎* Letters, stating particulars and lowest price, carriage free, to be sent to Mr. Bell, Publisher of "NOTES AND QUERIES," 186. Fleet Street. Notices to Correspondents. Replies Received.—Eagles' Feathers—Many Children—Longevity— Oasis—Newton, Cicero, and Gravitation—Burial of Suicides— Warwickshire Ballad—Algernon Sydney—Mother Damnable—Passage in Henry IV.—Moon and her Influences—Emaciated Monumental Effigies—Cane Decane—Hoax on Sir Walter Scott—Poison—Whipping Boys—Monument of Mary Queen of Scots—Portrait of Earl of Peterborough—Can Bishops vacate their Sees, &c.—Burials in Fields —The Three Estates of the Realm—Bawdricks for Bells—The Sclaters —St. Christopher—Arms of Thompson—Wyned—Lines on Crawfurd of Kilbirnie—Silent Woman—A Man his own Grandfather—Palæologus —Lines on a Bed—Inveni Portum, &c., and many others, which we will acknowledge in our next Number. A. B., who asks the meaning of Mosaic, is referred to our 3rd Vol., pp. 389. 469. 521. C. C. G., who asks the origin of "God tempers the wind," is referred to our 1st Vol., pp. 211. 236. 325. 357. 418., where he will
- 77. find that it is derived from the French proverb quoted by Gruter in 1611, "A brebis pres tondue, Dieu luy mesure le vent". Polynesian Languages. If Eblanensis will call on the Assistant Foreign Secretary of the Bible Society, he will be assisted in procuring the Samoan text, and such others as have been published. The Feejeean is just about to be reprinted, the first edition being out of print. Keseph's Bible. The Query on this subject from "The Editor of the Chronological New Table" has been accidentally omitted. It shall be inserted in our next Number. J. M. G. C. is thanked. His suggestions and communication shall not be lost sight of. Balliolensis is requested to say how a letter may be addressed to him.
- 78. SPECIMENS OF TILE PAVEMENTS. DRAWN FROM EXISTING AUTHORITIES BY HENRY SHAW, F.S.A. Although some few examples of the original designs, and many separate patterns taken from the scattered remains of these most interesting Pavements, are figured in divers Architectural and Archæological Publications; it is presumed, that if a series of specimens of the many varieties of general arrangement to be found in those still existing, together with a selection of the particular Tiles of each period, the most remarkable for the elegance and beauty of the foliage and other devices impressed upon them, were classed chronologically, and brought within the compass of a single volume, it would prove highly valuable as a work of reference; not only to architects, but to all who are engaged in furnishing designs for any kind of material where symmetrical arrangements or tasteful diaperings are required. The present work is intended to supply such a desideratum. It will be completed in Ten Monthly Parts. Each Part to contain Five Plates, royal 4to. printed in Colours. Price 5s. A Preface and Description of the various Pavements will be given with the last Number. No. I. was published on the 1st of May, 1852. Works by Mr. Shaw. DRESSES AND DECORATIONS OF THE MIDDLE AGES. In 2 vols. coloured, imperial 8vo. price 7l. 7s.; or on imperial 4to. the plates more highly finished and heightened with gold, price 18l. ILLUMINATED ORNAMENTS. From the sixth to the seventeenth century. Selected from Manuscripts and early printed books, carefully
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