The Geometry of Domains in Space (Birkhäuser Advanced Texts)

0'5:

O}

(1.3) (1.4) (1.5)

= {z : I(x) > OJ.

Notation 1.1.6 Throughout this book, we will use B(x, r), r > 0, to den.ote the open ball about x with radius r. We will also use iii(x, r) r ~ O,to denote the closed ball about x with radius r. Proof: We use the characterization that the function I is lower-semicontinuous if and only if {x : lex) > o} is open for every a e III By redefining I to equal 1 on {x: I(x) > 1}, we may and shall assume that / takes only finite values. E~hausting {x: I(x) > O} by compact subsets, we can find a sequence of points Plo P2, ... and positive reals U1, U2, ••• such that 00

{x: f(x) > O} =

U lR(Pi,Ui). i=l

For each i = 1,2, ... , define the function
Noting that

1,6[-1,1)

= ~ l(Pi) 1,6[-1,1) (Ix -

Pil 2 fun·

is everywhere less than 1, we see that each tPi satisfies

0'5: tPi :'5: I· For each i = 1,2, ... , set ki

{I I:

= sup ':~i a is a multi-index with lal '5:

Then the series 00

i} .

2-;

L 1 +k. tPi i=l

•

converges unifonnly, as do all its term-by-term derivatives of any order. The function defined by the series has the desired properties. • Proof of the Coo Varieties Theorem 1.1.4: Let V be a closed subset of lllN. We apply Lemma 1.1.5 with I = Xcv, the characteristic function of the complement of V. The resulting function

•

CHAPTER 1. ELEMENTARY TOPICS

6

Notation 1.1.7 Let A be a set in a topological space. We will denote the interior 0/ A by ..4, the closure 0/ A by A, and the boundary 0/ A by oA. Proof of the Coo Urysohn's Lemma 1.1.3: Let A and B be disjoint closed subsets of JRH. First construct a closed set A with

(AT, AnB=0,

Ac

a task we leave to the reader. Use the Coo Varieties Theorem 1.1.4 to obtain a non-negative Coo function "" with A = {x : 1/1(x) = O}. Set c = inf",eB 1/1(x). Apply the LSC Domination Principle 1.1.5 with

/(x )-_{Oc

E

if x A ifx¢A

to obtain a function t/J E Coo that has A as its zero set and is less than or equal to "" on B. We can then define I)!(x)

= {~f:~ o

if x ¢ A if x E A.

The function lit is Coo because 1/1 vanishes in a neighborhood of A. Also lit equals 0 on A and is greater than or equal to 1 on B. To complete the proof we compose I)! with any non-negative Coo function from R to R that vanishes on (-00,0] and equals 1 on [1,00), for example 1

~ JR '/'[0,1)

jt

4>[0,1)(8)

-00

ds.

•

Now let us look at Coo partitions of unity. Let E C RN be a closed set. Let U = {U.. } .. eA be an open covering of U. The following result will be proved in Section 3.5. Lebesgue Covering Theorem There is an open refinement V = {Vp} of the covering U (that is, each Vp E V is open and is a subset of some U.. E U) such that (i) V still covers E.

+ 1 ofthe sets Vp. (This assertion is summarized by saying that the covering V has valence not exceeding N + 1.)

(ii) No point of E lies in more than N

Now we have

1.1. SMOOTH FUNCTIONS

1

=

Theorem 1.1.S (Smooth partitions of unity) Let U {Ua}aEA be an open cover of a closed set E. Then there exist Coo functiorJs ,pj, each ,pj compactly.supported in some Ua/, such. that

Et/lj(Z)

=1

forall

Z

e E.

j

Further, no point of E lies in the support of more than N + 1 of the functions t/lj. Definition 1.1.9 The collection 0/ functions {t/lj} is termed a smooth partition of unity subotdinate to the covering U. Proof of Theorem 1.1.8: Use Lebesgue's Covering Theorem to pass to an open refinement V of E having valence not exceeding N + 1. We can also assume that V is countable (either by writing E as a countable union of compact sets or by appealing to Linde1of's Theorem, see for instance Dugundji [I], p. 174). For each j, apply Theorem 1.1.4 to the closed set cll; to obtain a Coo function p,j that is supported precisely on ll;. Replacing p,j by Jl.~ if necessary, but without changing notation, we can assume each p,j is also a non-negative function. Notice that each point of E lies in at most N + 1 of the sets ll; and that for each Z E E there is at least one j such that p'j(z) > o. Apply Theorem 1.1.4 to E itself to obtain a function p,* supported exactly on cE and which we may also assume is non-negative. Define

t/J(Z)

= p,*(z) + L>j(z). -

j

For each Z there is a neighborhood W", such that at most N + 2 of the functions involved in the definition of t/J are non-vanishing on W",. It follows that t/J is Coo. Note that t/J is nowhere zero, and we have

t/J(Z)

= :~::>j(x)

for Z E E.

j

Thus the functions

have the desired properties.

t/l.(x) = P,j(Z) 3 - ""(Z)

•

CHAPTER 1. ELEMENTARY TOPICS

8

1.2

The Concept of Defining Function

Definition 1.2.1 Let n c aN be a bounded domain; "Let p : IllN -+ III be a C le function, I: ~ 1. We say that p is a deflping function for n if the following propertiu obtain:

n = {x e aN : p(x} < O};

en = {x e aN :p(x} > O}; gradp(x)

=Vp(x) j

0 for all x E an.

(1.6) (1.7) (1.8)

Notice that, from freshman calculus, the vector field V p is normal to

an at each point of an. We often denote this vector field by v and its value at a particular boundary point P by v(P) or vp. Observe that, according to our definition of a defining function, vp will be an outward normal to 8n at P. Thus our setup automatically orients an, the boundary of the domain n. Definition 1.2.2 A vector v = (vt, ... ,VN) e IllN is a tangent vector to at P e 8n if v .L V pep), where the orthogonality is detern&ined by the usual Euclidean inner product. Another way to say this is that v. is tangent to an at P if

8n

N

'L)8p/OXi}(P} . Vi = O.

(1.9)

i=1

We write v e Tan(on, P). In some contexts it is useful to replace the defining function p with the function p(x) == p(x)/IVp(x)l. This function is well-defined whe~ x is near an and can be suitably modified away from 8n so that it, too, is a defining function. The attractive feature of p is that IVp(P)1 = 1 for every Peon. The process of replacing a given defining function p by another defining function p raises the question of whether the geometric objects that we wish to study in this book are dependent on the choice of defining function. Fortunately they are not. A significant part of what we do in the present section is to establish this independence. If keN and p is a Cle function that is a defining function for a domain n C IllN, then we say that n has cle boundary. One might wonder whether other reasonable definitions of Cle boundary are equivalent to this one. What if the boundary is locally the graph of a Cle function? What if the boundary is given parametrically? What if, instead, we wish to think of the boundary as a regularly imbedded Cle manifold? Fortunately, all of these approaches are equivalent. We shall see the details of this claim in what follows. Let us repeat that the condition defining tangent vectors simply mandates that w .L Vp at P. After all, we know from calculus that V p, which is parallel to Vp, points in a normal direction, and since the surface 8n

1.2. THE CONCEPT OF DEFINING FUNCTION

9

is (N - I)-dimensional the normal direction is uniquely determined; thus the normal direction, and hence the tangent vectors, are independent of the choice of p. In principle, this settles tbeissue of whether the tangent vectors to an are well-defined. Nevertheless, this point is so important, and the point of view that we are considering so pervasive, that further discussion is warranted. The issue is this: H p is another defining function for n, then it should give the same tangent vectors as p at any point P E an. The key to seeing that this is so is to write p(x) = hex) . p(x}, for h a function that is non-vanishing near an. Then, for P E an, N

'f)ap/aXj)(p) . Wj

.w,) (~(ah/IIz')(P). w,)

= h(P)· (t,(ap/IIz,}(p)

j=1

+ pep) . =

h(P)·

(t,

+

(ap/IIz, )(P) .

0,

Wi) (1.10)

because pep) = O. Thus w is a tangent vector at"P vis-a-vis p if a,r;,d only if w is a tangent vector vis-a~vis p. But why does h exist? After a change of coordinates, it is enough to assume that we are dealing with a piece of an that is a portion of fiat, (N - I)-dimensional real hypersurface Oust use the Implicit Function Theorem-see for example, Smith [1], Chapter .10, Theorem 8.2). That is, we may assume that the defining function p is given by . P(XbX2, ••• ,XN)

If Pis another defining function for hex)

={

= XN.

an near P, then we set

z~p(x)

if XN ~ 0,

(aptaXN )(x)

otherwise.

(1.11)

The Mean Value Theorem allows us to conclude that h is continuous along = OJ. The continuity of h and the vanishing of p on an is sufficient for the product rule to hold for the partial derivatives of p = h . p (even if h is not differentiable at points of an). Thus (UO) is verified. It is natural to ask just how smooth the function h defined by (1.11) must be. Because of the presence of ..J!I!...88in its definition, the best that can ZN be hoped is that h would be C"-1, and that is in fact the case. Applying the generalized Leibniz's Rule (1.1) to

an = {x : XN

x~p(x),

CHAPTER 1. ELEMENTARY TOPICS

10

we see that, for any multi-index a with then

lal $

(1)

aN . aa ____ - = ~(-1)' aN·I ax a XN P ~ (aN - i)!

k-l, if we write a = (a', aN),

(1 )i+l _ XN

a(a',aN-i)p

.. (1.12)

ax(a',aN-i)·

We can evaluate the limit of

(X1N)i+I

a(a',aN-i)p ax(a',aN-i)

as XN approaches 0 by using l'Hopital's Rule. The resulting limit is 1 (i + 1) i!

a( a'

,OIN -i)

P

ax(a',1+aN)·

Substituting this limit in the right-hand side of (1.12) and using the identity j~I = E{=o (~r we see that the limit as XN approaches 0 of the righthand side of (1.12) is 1 a(O/,1+ a N)p (1.13) 1 + aN ax(a' ,1+aN) .

((),

The discussion above proves the following: Proposition 1.2.3 If p, P are C le defining junctions for a domain there is a C le - I function h defined near an such that p = h· p.

n,

then

The result in Proposition 1.2.3 is sharp. In fact, the next lemma shows that if one defining function for the domain n is P(XI,X2, ... ,XN) =XN

and a second defining function, near the origin, is P(XI,X2, ... ,XN) = xN(1 + Ixl k - l +>.),

with 0 < A < 1, then pECk while h = PiP E CIc-1 \ Ck. Lemma 1.2.4 Let P(XI, X2, ... , XN) ~ 0 be a homogeneous polynomial of degree d, let l be an integer, and let A satisfy 0 < ,\ < 1. If we set ~(x) = P(x) Ixll+,\

'I. CO if d + l $ -1. Proof: The fact that d + l $ -1 implies ~ 'I. CO is seen immediately by using polar coordinates, so we may and shall assume that d + l ? o. Since P ~ 0, we can find "I = ("11, "12, • .• , 'YN) such that P("I) t= o. Then then ~ E Cd+i \ Cd+l+ I if d + l ? 0 and ~

we see that

1.2. THE CONCEPT OF DEFINING FUNCTION

11

which is clearly not in Cd+l+l, so we conclude that C) ;. Cd+l+l. Using polar coordinates again, we see that if d + l 0, then C) E CO. The rest of the lemma then follows by induction on d+l using the fact that

=

88Xi \x\m

= m Xi \x\m-2,

which allows us to conclude the following: A sum of functions of the form

E• Pi(~) Ixl'l+>',

(1.14)

i=1

where each 1'; is a homogeneous polynomial of degree di , each integer, and D = d 1 + 11 = d2 + i2 = ... = d. + i.

li

is an

has as its partial derivatives functions also of the form (1.14), but with the value of D reduced by 1. • This somewhat protracted discussion of a small technical point seems necessary because it is recorded incorrectly in many places in the literature. Defining Functions with Unit Normal Vectors

Note that, while the ratio of any two C k defining functions for a domain must be at least C k - 1 , it is not the case that if a given function p is a C k defining function for some domain, then it can be multiplied by any C k - 1 function and have the resulting product be a C k defining function. Generally, such a product can only "be expected to be of class C le - I . In particular, the defining function p defined by p = pll"pi is one order of differentiability less smooth than p, and that is the price to" be paid in obtaining a unit normal vector by this change of defining function. It is not necessary to sacrifice an order of differentiability to obtain a defining function with a unit length gradient vector. Provided the domain has a boundary that is C 2 or smoother, the signed distance to the boundary will do the job. This is not obvious; it is a somewhat subtle application of the Inlplicit FUnction Theorem as we will see below. Lemma 1.2.5 If n has a C k k ;?: 2, defining function p and P E an, then there is a neighborhood U of P such that for all Q E U there is a unique point Q' E an nU with

IQ - Q'I = dist(Q,ao).

Moreover, Q' is a Cle -

1

(1.15)

function of Q.

Proof: For fixed Q, the point Q' must be an extremum of the function IQ - Q'1 2 subject to the constraint p(Q') O. By the Lagrange Multiplier

=

12

CHAPTER 1. ELEMENTARY TOPICS

Theorem (see for example Smith [1], Chapter 11, Theorem 4.7) there mUst exist a real number >. such that Q' satisfies Q - Q' - >. Vp{Q') p(Q') = 0.

= 0,

(1.16)

We will apply the Implicit Function Theorem to show that for Q near enough to P the equations (1.16) have a unique solution near P that is a Ck-l function of Q. Consider the function F : Jll2N+l --t aN+! defined by

( 8,>. ) ~ .(.

Q - Q' - >. V p(Q') ) p(Q') .

The derivative (or Jacobian matrix) DF of F is represented by the matrix of size (N + 1) x (2N + 1) given by -INxN - >.Hess[p(Q')]

·[Vp(Q,)]t where Iixt denotes the l x l identity matrix and Olxm denotes the f. x zero matrix. So we have

m

INxN

DF(P,P,O) = (

[Vp(P)]t

OlxN

Note that F(P, P,O) = apply if

°

E ]RN+l.

.. The Implicit Function Theorem will

Vp(P)

[V p(P)]t

o

1#0,

but this is true since one can readily verify by induction on l that, for VERi,

-Itxt

det [ vt

: 1~ (-1)'lvl'·

•

Theorem 1.2.6 If n has a C k , k :2: 2, defining function p, then the signed distance function p(Q) = sgn(p(Q)) dist(Q,an) is a C k function in a neighborhood of on.

1.2. THE CONCEPT OF DEFINING FUNCTION

13

Proof: Fix a point Peon. Let us introduce the notation 9 : U -. JRN for the function mapping Q to Q', the nearest point of 00, as in the preceding lemma. Equations (1.16) then become

g(%) = ~(x) ('\1 p) pog(%) = O.

%-

0

g(x)

(1.17) (1.18)

It follows that also

=.\(%) I(Vp) g(x)1 N p2(%) = 1% - g(x)12 = ~)Xi - gi(x»2. p(%)

0

(1.19) (1.20)

i=1

Differentiating (1.18) with respect to N a L....!!....

Xj,

g(x)

0

i=1 aXi

we have

a.

....!!!.. == 0,

(1.21)

aXj

for j = 1,2, ... , N. Differentiating (1.20) with respect to %j and using (1.17) and (1.21), we find that N

=

2L (6

ij -

;:~)

i=1

=

..

(Xi -

1

N

2(Xj _ gj) _ " " agi

L..J ax·1

~

__i=1

=

2(Xj - gj)

=

2;\-8 og.

gi) ap

8x·•

0

g(x)

8p

Xj

Thus we have

pVP=>.(Vp)og. By (1.19) we conclude that

V- _ (Vp) og P-I(Vp)ogl

(1.22)

must hold, at least on the complement of a~. But the left-hand side of (1.22) is continuous and the right-hand side is le - 1 , so the equation must hold on 80 as well. This shows that V p is C Ie - 1 • Therefore, pis Cle .•

c

Other proofs that the signed distance function is locally CIe, k ;::: 2, if the domain is Cle can be found in Foote [1], Gilbarg and Trudinger [1], and Krantz and Parks [1].

CHAPTER 1. ELEMENTARY TOPICS

14

Equivalent Notions of Smooth Boundaries Let us now turn to the idea of the boundary being locally a graph. Fix a domain 0 C ]RN and a point P e ao. We say that ao is locally the graph of a function near P if there is an open N-dimensional neighborhood U of P, a Euclidean coordinate system tl, t2, ... , tN on U, and a function U(tlo t2,"" tN-I) such that

ao n U = {(tlot2,""

tN-I, U(tlo t2,"" tN-I»

e U}.

This definition is most convenient if the set U is a product region in the t-coordinate system. Observe that if·aO is locally the graph of a C k function u near P, and if the positive tN axis points out of the domain, then the function pu(t) == tN - U(tI, t2, ... , tN-I) behaves like a defining function for 0 near P. In other words, if we express pu in the original spatial coordinates XI,X2, .•. ,XN on U, then pu(x) > 0 on Un cO, pu(x) < 0 on uno, and "pu =I 0 on ao. If 80 is locally the graph of a C k function near every point of 80, then we cover 80 by a family of open sets Uor , each with its associated defining function Per. Now apply the Partition of Unity Theorem 1.1.8 to this open covering of the set E = 80. We obtain compactly supported, Coo functions ,pj that sum to 1 on 80. Setting 00

p(x)

= 2: ,pj(x)Porj (x) .. j=I

where ,pj is supported on Uer" we obtain a function defined on a full neighborhood of 80 that satisfies the properties of a defining function on that neighborhood. It is then elementary to extend the defining function to'all of space if needed. The preceding machinery may be run in reverse. If p is a C k defining function for 0, then fix a point P E ao. Assume the coordinates have been rotated so that Vp points in the direction of the positive XN axis. The Implicit Function Theorem then implies that there is a neighborhood V of ~ (PI. P2 , ••• ,PN-J) and a Ck function u(xt, X2,···, xN-d such that

on V. But this just says that 80 is locally the graph of the function u near P. Another common method for specifying a hypersurface in space is parametrically. Let W C IRN - 1 be an open set and let tPt,'ljJ2, ... ,'ljJN be C", real-valued functions on W. Set t' = (tl, t2,'" , tN-I) for the coordinate on W. If the mapping

15

1.3. MEASURE THEORY

is one-to-one and if the matrix (IJ'IjJ,/XIe) has rank N - 1 at each point of W, then the image M of ~ is called a Cle hypersurfacegiven parametrically. Now let us relate the notion of parametrically given hypersurface to some of the preceding ideas. With notation as in the last paragraph, fix a point P = (PI, P2, ... , PN) in the surface M. The matrix

(~~

!!!I!J.

...

~

...

~

~

... ..P!I!JL. atN-l

8tt

tlt2

at,

Jlt~l ~)

(1.23)

has rank N - 1. For convenience of notation, let us suppose that the submatrix consisting of the first N - 1 rows of (1.23) is non-singular. Then the Inverse Function Theorem tells us that the mapping

has a C k inverse function near (Pt,P2 , ••• ,PN-t). We will denote that inverse function by +. Then near P we can write M as the graph of a function: (Xt, X2,··· ,XN-t, 'ljJN[+(XI,X2, ... , XN-I)]), (Xt,X2, ... ,XN-t) near (Pt,P2, ... ,PN-I). Thus we have a third method for thinking of the boundary of a domain: as a locally parametrized hypersurface. We see that all three points of view are equivalent, and one passes back and forth among them using local coordinates and the Implicit Function -'fheorem. A fourth point of view is to Consider the boundary as a locally regularly imbedded Ck manifold (see Hirsch [1]). Since that method will not be used in this book, we shall not discuss it in detail.

1.3

Measure Theory

In this section we review some basic measure theory, mainly to set our notation and terminology rather than to provide a complete treatment. The reader interested in an exhaustive presentation should see Federer [4]. Definition 1.3.1 Let X be a set and let S be the set of all subsets of X. By a measure on a set X we mean a function m:

satisfying the conditions

(i) m(0) = 0,

S -+ [0,+00]

CHAPTER 1. ELEMENTARY TOPICS

16

L m(B) if B is at most countable and A CUB.

(ii) m(A) ~

BE8

BE8

IT m is a measure on a set X, it is convenient to postulate, as we have, that every subset A C X has associated a wlue m(A), but some subsets are more important than others. Definition 1.3.2 Let m be a meanre on the set X. A subset A C X is said to be m-measurable if

m(B) = m(BnA)+m(B \A)

(1.24)

holds for every Be X. It is obvious from this definition (and the definition of a measure) that any set having measure 0 is m-measurable. The sets of measure 0 are negligible in measure theory, and any statement S(x) containing a free variable from a set X with a measure m is said to hold for m-alrnost all x or m-almost everywhere if m( {x eX: Sex) is faIse }) o.

=

The main result concerning the m-measurable sets is expressed by the following definition and theorem (see Federer [4], §2.1.3). Definition 1.3.3 By a u-algebra in a set X we mean a family S of subsets ·of X such that ..

(i) 0 e S, (ii) if A

e S,

XeS,

then cA

e S,

(iii) if Be S is at most countable, then UBE8 Band ofS.

nBEB B are elemen"ts

Theorem 1.3.4 lfm is a measure on X, then the family ofm-measurable sets forms a u-algebra in X and m is countably additive on pairwise disjoint collections of m-measurable sets. Note that many texts would use the term outer measure for what we have called a measure. Because of the preceding theorem, if m is a measure in our sense, then when restricted to the u-aIgebra of m-measurable sets, m is a measure in the alternative sense of a function defined on au-algebra and countably additive on disjoint unions.

=

Example 1.3.5 Let X R. For any non-empty open interval (a, b) set m(a b») = { b - a ~f - 00 < a < b < +00, , +00 If a = -00 or b = +00.

c IR,

Since any open subset of III is uniquely expressible as an at most countable union of pairwise disjoint intervals, m will be well-defined on open sets if it

1.3. MEASURE THEORY

17

is extended additively to pairwise disjoint unions of open intervals. Finally, for an arbitrary A C IR, we define

meA)

= inf{m(V) : V

is open and

A c V}.

We leave it to the reader to verify that this defines a measure. This measure is called Lebesgue Dleasure and will be denoted by l.. Sets that are l.-measurable are referred to as Lebesgue measurable sets. Measure theory becomes more interesting for the analy.st when there is also a topology on the space. Recall that, if X is a topological space, then the Borel sets are the elements of the smallest O'-a1gebra that contains the open sets.

Definition 1.3.6 (i) A measure m is regular if every set A is a subset of a measumble set B having the same measure, that is m(A) = m(B).

(ii) Let X be a topological space and let m be a measure" on X. We say that m is Borel regular if all Borel sets are m-measumble and for every subset A C X there exits a Borel set B with "

A

c

Band m(A)

= m(B).

(1.25)

(iii) Let X be a locally compact Hausdorff space and let m be a measure on X. We say that m is a Radon measure if all Borel sets are mmeasumble and the following conditions hold:

• IlK C X is compact, then m(K) < 00. • If V C X is open, then m(V) = sup{m(K) : K is compact and K C V}. • For any A eX, m(A)

= inf{m(V): V

is open and A C V}.

The interested reader should consult Federer [4], §2.2.4, to see the proof that any sufficiently rich Borel regular measure on a complete separable metric space must have a non-measurable set. The proof in Federer [4] uses the Axiom of Choice in the form of the Well-Ordering Principle. In general, the question of measurability is inextricably tied to the study of abstract set theory. In a notable paper, R. M. Solovay [1] has shown that if there exists an inaccessible cardinal, then there is a model of set theory in which all subsets of III are Lebesgue measurable. Obviously, one would now want to know whether or not an inaccessible cardinal exists, but unfortunately it is impossible to show that the existence of an inaccessible cardinal is consistent with the usual axioms of set theory (see Jech [1], Theorem 27, p.85).

CHAPTER 1. ELEMENTARY TOPICS

18

Proposition 1.3.7 (Inner Regularity of Radon Measures) Suppose m is a Radon measure on X. If A is m-measumble, m(A} < 00, and o< E, then there exists a compact set K C A such that m(A} < m(K} + E. Proof: There exists an open set" U such that A C U and m(U \ A} < E/2. With the choice of U now fixed, there exists another open set V such that U\A C V and m(V} < E/2. We can also find a compact set K' with K' C U and m(U \ K') < E/2. Finally, we set K = K' \ V, so K is compact, K C A, and we have .

m(A \ K) = m(A \ (K' \ V» 5 m(U \ K') +m(V}

< E.

•

The following result is a useful tool (see Federer [4], §2.3.2(9}). Theorem 1.3.8 (Caratheodory's Criterion) Ifm is a measure on the metric space X, then all Borel subsets of X are m-measurable if and only if m(A U B} ~ m(A} + m(B) (1.26)

holds whenever A, B C X with dist(A, B)

> O.

Suslin Sets Caratheodory's Criterion shows us that all Borel subsets of]R are Lebesgue measurable. It is natural to ask whether the Borel sets are the only Lebesgue measurable sets. Answering that and related questions leads one into the study of descriptive set theory and away from the topics in this book. We shall only briefly touch on these matters. Definition 1.3.9 A set A C ]RM will be called a Sustin set if there exists a Borel set B C jRM X jRN, for some N, such that A is the image of B under projection onto the first factor. Originally, Suslin sets were known as analytic sets 1 (and still are known as ensembles analytiques in French). The classic book on the subject is Lusin [1].2 The main facts are contained in the following theorem (see Federer [4], §2.2.1C}-2.2.13).

1Folklore has it that as a graduate student Suslin was given the task of proofreading a paper by the already famous Lebesgue in which Lebesgue had stated as a lemma, the proof of which was omitted because it was obvious, the assertion that the projection of a Borel set is a Borel set. Suslin realized that the assertion was in fact false and produced a counterexample. Jech [1], page 593, confirms at least the broad outline of this story and cites the original papers. 2The author's name is also transliterated as Luzin.

1.3. MEASURE THEORY

19

Theorem 1.3.10 3 (i) Every Borel set is a Suslin set:

(ii) II A is /I Suslin set in I(A) is /I SusUn set. (iii) There esist6

/I

RMI

/lnd

f : A -+ JlM2 is continuous, then

Suslin set that is not

/I

Borel set.

(iv) II m is /I meosure on aM such that all Borel sets are m-meG8uroble, then every Sulin set is m-measumble.

Measurable Functions Definition 1.3.11 Suppose thahn, is a measure on X and th/lt Y is a topological space. A function defined on m-almost all 01 X and having values in Y is said to be m-measurable ill-I(U) is m-measuroble for every oPen u c Y. A pair 01 m-measumble functions I and g are considered to be. equal (for purposes 01 measure theory) il they agree at m-almost all points of X. It is a series of elementary exercises to verify that the set of real-valued . (or complex-valued or]RN -valued) m~asurable functiollS is closed under the usual algebraic operations--of course, due care must be exercised in dealing . with division by zero. It is another elementary exercise to verify that, for measurable functions taking values in a metric space, the pointwise almost . everywhere limit of a sequence of measurable functions is measurable. While the condition of measurability is not very restrictive, it is still sufficient to ensure some nice properties, as the next two theorems show . (see Federer [4], §2.3.5 and §2.3.7).

Theorem 1.3.12 (Lusin's Theorem) If m is a Radon measure on a locally compact Hausdorff space X, f : X -+ ]R is m-measuroble, A is a compact set with meA) < 00 <, and 0 < E, then there exists a compact set C such that meA \ C) < E and f restricted to C is continuous. Theorem 1.3.13 (Egoroff's Theorem) If m is a measure on X with m(X) < 00, the m-meG8umble functions Ii : X -+ 1R, i = 1,2, ... , converge m-almost everywhere to g : X -+ 1R, and 0 < E, then there exists an mmeasurable set B such that m(X \ B) < E and the convergence of Ii to g is uniform on B. 3The reader comparing our definition to that in Federer (4), §2.2.10, should note that Federer's space N is homeomorphica1ly mapped onto a Borel subset of]R by the mapping 00

(01.n2 •... ) .....

L exp (- E~=l nj). i=l

CHAPTER 1. ELEMENTARY TOPICS

20

One of the main reasons for the invention of measure theory is the extension of the notion of integration from the relatively simple Riemann integration taught in freshman calculus to a more general and flexible tool useful in modern analysis. . Definition 1.3.14 Let m be a measure on X. (i) By an m-step function we mean an m-meosurable function, ing at most countably many real valuell such that

II,

tak-

(1.27) is well-defined, using the convention 0 . 00 = 00 • 0 = o. That is, the series in (1.~7) either converyes, diveryes to +00, or diverges to''':'oo.

(ii) If f is an m-measurable function defined on m-almost all of X and taking values in [-00, +00], we define the upper integral of f to be the infimum of the expressions in (1.27) as s varies over all step functions that are greater than or equal to f at m-almost all points of x. The upper integral of f is denoted by

(iii) Likewise, if f is an m-measurable function defined on m-almost all of X and taking values in [-00, +00], we define the lower integral of f to be the supremum of the expressions in (1.27) as s varies over all step junctions that are less than or equal to f at m-almost all points of x. The lower integral of f is denoted by

l

fdm .

r

(iv) If f dm = I. f dm, then we denote the common value of the upper and lower integrals by

J

fdm.

(v) We say that f is m-integrable if the upper and lower integrals of f are equal and finite. Often the measure m is clear from context and we will say simply that f is integrable

(vi) Suppose X is a topological space and f is an m-measurable junction on X. If for each p E X there exists an open set U with p E U and .um that f . Xu is m-integrable, then we say f is locally m-integrable or locally integrable when the measure is clear from the context.

21

1.3. MEASURE THEORY

The preceding integration of m-measurable functions behaves well with respect to limits as the next three classical theorems tell us.

Theorem 1.3.15 (Fatou's Lemma) Suppose m is a measure on X. If are non-negative m-measurable functions on X, then

h, /2, ...

li~ inf .-+00

li~ inf fi dm. lx .....

( fi dm $. (

lx

(1.28)

00

Theorem 1.3.16 (Lebesgue's Monotone Convergence Theorem) Suppose m is a measure on X. If h, /2, ... are non-negative m-measuroble functions on X such that holds for m-almost all x EX,

fa(x) $. fi+1(x) then

.lim ( /; dm

.-+00

1x

= 1(x ..... .lim h dm. 00

(1.29)

Theorem 1.3.17 (Lebesgue's Dominated Convergence Theorem) Suppose m is a measure on X. If h, /2, ... and g m-measuroble functions on X such that /;(x). converges for m-almost all x E X,

Ih(x)1 $. g(x) holds for m-almost all x

E

X,

and

then .lim { /;dm

.....oolx

=-r

.Jim hdm.

lx .....

00

(1.30)

.

Product Measures and Fubini's TheoreIn Definition 1.3.18 Suppose m is a measure on X and n is a measure on Y. The product measure m x n is the measure on X x Y defined for SeX x Y by setting (m x n)(S) equal to the infimum of the set of numbers of the form 00

E m(Ai) . nCB;), ;=1

where A 1, A 2 , ••• are m-measurable subsets of X, B 1, B 2 , ••• are n-measuroble subsets of Y, and (1.31) i=l

In (1.31) we use the convention that 0·00

= 00 . 0 = o.

CHAPTER 1. ELEMENTARY TOPICS

22

The reader can find the proof of the following theorem in Federer [4], §2.6.2.

Theorem 1.3.19 (Fnbini) Suppose m is a measure on X and n is a measure on Y. (i) m x n is a regular measure on X x Y.

(ii) The cartesian product 0/ measurable sets is m x n measurable, and the measure oj a cartesian product is the product oj the measures. (iii) IJ SeX x Y is a countable union oj sets oj finite measure, then (a) {x: (X,II) E S} is m-measurable Jor n-almost every II E Y, (b) {II: (x,y) E S} is n-measurable lor m-almost every x E X, (c)

(m x n}(S)

! = ! =

m({x: (x,y) E S})dn n({y: (x,y) E S})dm.

(iv) 1/ J is an.m x n integroble /unction such that {(x, y) : /(x, y) =F O} is a countable union 0/ sets 0/ finite measure, then .

! I d(~

x n}

=

! (// ~m)

dn =

! (!

J

d~) dm.

Definition 1.3.20 Lebesgue measure on]RN will be denoted by eN and is defined inductively by setting . £N

=£ X eN-l.

Integrals with respect to Lebesgue measure will often be written using the notation

Lebesgue Spaces Definition 1.3.21 Let m be a measure on X. Suppose 0 < p < measurable J : X ~ C we define

II/lip =

Ix IflP

dm.

00.

For a

(1.32)

The Lebesgue space V(m) is the set of measurable f : X ~ C for which IIfllp < 00. We call II' lip the V-Dorm. In case X =]RN and m =£ N, it is customary to write V (]RN ), instead of IJ'(£N}.

1.3. MEASURE THEORY

23

In the preceding definition, the value p = 00 is conspicuously absent. A separate definition is required:

Definition 1.3.22 Let m be a measure on X. For a measurable I : X -+ C we define

Illlico = inf {t : 0 < t < 00

and m({z: t < I/(z)1})

= O}.

(1.33)

We set LCO(m) equal to the lamily 01 measurable I : X -+ C lor which we call II· lleo the Leo-norm, and in case X = JRN, m = CN, we write LCO(JRN) instead 01 Leo (C N ).

llllIco < 00,

For future reference we state some of the classic results concerning the Lebesgue spaces. Theorem 1.3.23 Let m be a measure OR X. For 1 $ p $ 00, the Lebesgue space V'(m) is a complete metric space with the distance between I and g given by III - gllp· Theorem 1.3.24 Let X be a locally compact Hausdorff space and let m be a Radon measure on X. For 1 $ p < 00, the set of continuous functions with compact support Cc(X) is dense in the Lebesgue space V(m) . . For 1 < p < 00, it is not obvious that the function d(J, g) = III - gllp on V(m) satisfies the triangle inequality and thus defines a metric. The underlying result is the following: Theorem 1.3.25 (Minkowski's Inequality) Let m be a measure on X and let I and g be m-measurable functions taking values in [0,00]. For 1 < p < 00, it holds that

(Ix

(J

+ g)Pdm) l/p $

(Ix IPdm)

l/p

+ (lxgPdm) l/ P .

(1.34)

Another of the important inequalities in analysis is that of Holder which applies to pairs of specially related exponents as in the next definition. Definition 1.3.26 Numbers p and q with 1 < p, q < gate exponents il 1 1

00

are called conju-

-+-=1. p

(1.35)

q

Theorem 1.3.27 (Holder's Inequality) Let m be a measure on X and let I and g be m-measurable functions taking values in [0,00]. II p and q are conjugate exponents with 1 < p, q < 00, then it holds that

Ix

Igdm $

(Ix I Pdm)

l/p

x

(Ix g9 dm)

l/q

(1.36)

CHAPTER 1. ELEMENTARY TOPICS

24

Definition 1.3.28 It is also customary to say that p and q are conjugate exponents ilp 1 and q 00 or i/p= 00 and q 1.

=

=

=

Smoothing

The goal of a smoothing operation is to achieve the ability to approximate the objects in a category of interest by objects that have better properties than generally are expected in the category. The prototype of such a theorem is the Weierstrass Approximation Theorem (see for example Krantz [2], Theorem 9.8) in which the category in question is that of continuous functions on a compact interval and the approximating objects are the polynomials. The impetus to develop smoothing operations is the hope that one will be able to extend proofs based on intuition concerning smooth objects to the entire category under consideration. The main tool used in smoothing is the operation of convolution: Definition 1.3.29 For Lebesgue measumble junctions I, 9 : JRN -+ JR the convolution 01 I and 9 is the junction I *9 given by

(/*g)(x) =

!

l(x-y}g(1!}dy1dY2 ... dYN

(1.37)

lor all x such that the integral is defined.

It is not immediately apparent what the domain of the convolution of f and 9 will be, nor what estimates the convolution might satisfy. It clearly follows from HOlder's Inequality (1.36) that if f E V(JRN } and 9 E Lq(JRN}, for a p~r of conjugate exponents P and q, then (f * g)(x) is defined for all x E JRN. The situation is more subtle than that however, and we have the following result as a consequence of Fubini's Theorem (see Rudin [2], Chapter 7): Theorem 1.3.30 II I,g E Ll(JRN ), then (f * g)(x) is defined lor almost every x E JRN and I * g E Ll(JRN ) with (1.38) To apply convolution to smoothing, recall the function ¢[-l,l) constructed in Section 1.1. The important properties are ¢[-l,l) ~ 0,

> 0 if and only if Ixl < 1, E COO(JR).

¢[-l,l)(X) ¢[-l,l)

(1.39) (lAO) (1.41)

By multiplying ¢[-l,l) by a constant, but without changing notation, we can insure the following additional property holds:

i:


= 1.

(1.42)

25

1.3. MEASURE THEORY

For each positive integer N and each u > 0, we define the function ""IT called a Friedrichs mollifler by setting. . .

""IT (Xl , ... ,XN) = u- N

N

II tPt-l,l](X,/U). i=l

Such a family of functions is used to produce smooth approximations to continuous functions. The main result on smoothing is the following (see Smith [1], Chapter 13, Section 12): Theorem 1.3.31 Let f : lllN ~ III be locally integrable.

(i) For each u

> 0, the function

is Coo.

(ii) supp (f * ""IT) C

{x:

dist (x, supp (f»

~ v'Nu}.

(iii) If f is in LP, then f *""IT converges to f in the. £P -norm as u decreases to O. (iv) If f is continuous and C is a compact set; then f uniformly to f as u decreases to O. (v) If f is uniformly continuous on "iN , then f to f os. u decreases to O.

* ""IT

* ""IT

converges

converges uniformly

Part (iii) is proved using the following generalized Minkowski Inequality (see Zygmund [1], Chapter I, Equation (9.12»: (1.43) which expresses in a very general way the fact that the norm of a sum is less than or equal to the sum of the norms.

Chapter 2

Domains with Smooth Boundaries 2.1

The Tangent Bundle and Normal Bundle of the Boundary

Notation 2.1.1 We will denote the sphere of radius r > 0 about x E ]RN -by S(x, r). The unit sphere in]RN will be denoted by SN-I so that SN-I -S(O, 1).

=

-Definition 2.1.2 Let p be a defining function for the domain n. The Gauss map is the function Q : on -+ SN-I defined by v:.p(x)" g(x) = IVp(x)1

(2.1)

for x E on. It is clear that Q extends via the same formula to a neighborhood of on. We will use the same notation 9 for the extended function. The components of the Gauss map will be denoted by

(2.2) Clearly, Q(x) is a unit vector normal to on at x E on. Note that if p, and thus n, is in smoothness class CIe, then 9 is in smoothness class C le - l . In general differential topology (see Hirsch [1] for example), the normal bundle of a submanifold of Euclidean space consists of all pairs (p, w) where p is a point of the submanifold and w is a normal vector to the submanifold at the point p. For the boundary of a domain in Euclidean space, the normal bundle has the following natural definition. Definition 2.1.3 Let n be a domain with defining function p E Cle, k ~ 1. The normal bundle of on, denoted Nor(on), is the subset of]RN x JRlV

27

28

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

consisting of the points ((z,t!1{x» : x E R.N with p{X) For x E an, Nor{an,x)

= 0,

t E R.}.

= {t!1{x) : t E R.}

is the normal space to an at x. H X{Ul,U2, .•• ,UN-l) with p{x) = 0, then

= x{u)

is a local C le parametrization of, the points

(x{u) , tg{x{u») is a local C Ie - 1 parametrization of the normal bundle. It is easy to see that, if the differential of the parametrization u ~ x{u) is ofrank N - 1, then the differential of the parametrization (u, t) ~ (x(u), t!1{x(u))) is of rank N. Recall that the tangent vectors to the boundary of a domain were defined in Definition 1.2.2. The following definition is consistent with that earlier usage. Definition 2.1.4 Let n be a domain' with defining function p E Cle, k ~ 1. The tangent bundle of an, denoted Tan{an), is the subset of lIlN x lIlN consisting of the points

{(x, v) : z E lIlN with p{x) =

0, v E R.N with V· !1{x) = OJ.

The tangent bundle of a manifold consists of pairs (p, v) where p is a point of the manifold and v is a tangent vector to the manifold at the point p. In an abstract manifold it is 'not assumed a priori that there is an imbedding into Euclidean space, so the tangent space must be described .in terms of differentiation operators. For the boundary of a domain, there is a natural set of N tangential differentiation operators associated with a 'given defining function. We will see that they span the {N -I)-dimensional space of tangential differentiation operators. Definition 2.1.5 Let n be a domain with defining function p E Cle, k ~ 1. For i = 1,2, ... , N, we define the linear differential operators

(2.3) That the operators bi are tangential differentiation operators on an is shown by the next lemma.

= {x: p{x) = O}

2.1. THE BUNDLES ASSOCIATED WITH A BOUNDARY

I and 9 are Cl functions on some

Lemma 2.1.6 Suppose Xo E 80 and neighborhood U 01 Xo . such that I(x) then

= g(x)

un an,

lor x E

oi/(xo) = Oig(XO), i

= 1,2, ... ,N.

Proof: Set h = I - 9 in U. Since h(x) = 0 for x E U Vh(xo) = Ag(X), for some A E R, that is,

Oh

~(xo)

VXi for i

29

n on,

we have

= AGi(XO),

= 1,2, ... , N: We compute

• The next proposition tells us that the set of Oi is (N - 1}-dimensional and . tells us which of the Oi form a basis. Proposition 2.1.7 For x such that V'p(x) "I 0, the set 01 differentiation operators {Ol.02, ... ,ON} is linearly dependent, while ilGio =I- 0, then (2.4)

are linearly independent.

Proof: Since Vp(x) "10, it follows that g(x) is defined. Not all Gi(X} are zero, but the linear combination

is easily seen to vanish. Thus {01, 02, ... , ON} is a linearly dependent set. To see that the second part of the proposition is true, we suppose for convenience that io = N, so GN 'I- o. IT N-l

0=

L

i=1

N-l

'1Oi

=L

i=1

{}

N-l

L

'1- {}Xi

N

CiG i

i=1

N-l

L CiG = o. i

i=1

{}

L Gj(x)-, {}x; j=1

30

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

But then E!~1 CiOi simplifies to just Ef!,~1 Ci 8~i. We conclude that Cl = ... = CN-l = 0. Thus Ol! 02, ... , ON are linearly independent. •

=

C2

In the next section· we will see the utility of the operators Oi in giving some simple formulas for the curvature of the boundary of a domain. The previous discussion has shown that the differentiation operators can be used to represent the abstract tangent space of the boundary of a domain. The correspondence to the more concrete Definition 2.1.4 is demonstrated by replacing the differentiatiop operators ~ in the definition of Oi by the standard basis vectors ei.

Definition 2.1.8 Let n be a domain with defining function p E C", k ;:: 1. We define the tangent vectors N

T;

= ei - Gi(x)

2: Gj(x)ej j=1

One easily checks that T; • g = 0,

for i

= 1,2, ... , N.

As in the proof of the preceding proposition, N

LG/T; = 0, i=l

and, if Gio

f. 0, then

are linearly independent. It may seem wrong to the reader to consider a set of N tangent vectors rather than N -1, but it is useful to have a set of vectors that always contains a basis, even as the point under consideration moves over 8n. We have the following theorem: Theorem 2.1. 9 Suppose n is a C", k ~ 1, domain in JRN with defining function p. Then the tangent bundle can be written

{(x, E!l viT;(x)) : x E JRN with p(x)

= 0,

v = (V1,V2, ... ,VN) E aN}.

If x(ur, u2, ... , UN-t} = x(u) is a local C" parametrization of the set of points with p(x) = 0, for u in a neighborhood of Uo, and if G N(X(UO)) f. 0, then N-1

(

x(u), ~ v/T;(x(u))

)

2.1. THE BUNDLES ASSOCIATED WITH A BOUNDARY

31

is a local C"-1 parametrization of the tangent bundle. It is easy to see that, if the differential of the parametrization u ~ z(u} is of rank N -1, then the differential of the parametrization (u,v) ~ (z(u) , EVi7i(X(U}}} is of rank 2(N -1). H GN(X(UO}) 0, then there is some io such that Gio(x(Uo» =F 0 and a similar parametrization is used in this case.

=

Alternative Definitions of Tangent Structures The preceding discussion of the tangent and normal bundles has been based on calculus, but one can give more geometric definitions such as the next.

Definition 2.1.10 For any set S C IRN and any point pES we define the tangent ·cone of S at p to be the set of v E aN such that for each f > 0 there ezist q E Sand r > 0 so that

Iq - pi < f,

Ir(q - p} -

vi < f.

We denote the tangent cone of S at p by Tan(S,p). It is easy to verify that if n is a Cl dom~ and p E an, then Tan(an,p) as defined by Definition 2.1.10 coincides with the set Tan(an,p} defined in Definition 1.2.2, so our use of the notation Tan(S,p) is justified. Other easy observations are the following: The tangent cone is a closed set. H v E Tan(S,p) and r ~ 0, then rv E Tan(S,p). Ifp E 5, then Tan(S,p} = IRn. It will also be useful to be able to speak of the possible "continuity" of the tangent cone Tan(S,p} as p varies. For this we need a topology. The Hausdorff distance (see Section A.I) defines a useful metric on the set of non-empty compact sets in Euclidean space. The Hausdorff distance between C1 and C2 will be denoted by HD(C},C2 ). We will use the Hausdorff distance to define a metric on the space of non-trivial closed cones, and thus to define a topology:

Definition 2.1.11 (i) A set C C

]RN

is a cone if

C

= {rx : 0 $

r <

00,

z E C n S(O, I)},

and C is a non-trivial cone if C n S(O, 1) =F 0. We do not require that a cone be closed under the operation of vector addition.

(ii) For C} and C2 non-trivial closed cones in aN we define the distance between them to be HD(C1 nS(0,1),C2 nS(0,1». Another way to look at the tangent cone is provided by the following result.

32

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

LelllIlla 2.1.12 For a set S C RN and a point pES, Tan(S,p) is the transl~tion to the origin of the union of all half-lines

r(p, q) := {p + t(q - p) : t ;:: O}

(2.5)

originating at p such that a sequence {qi} of points in S \ {p} exists with qi converging to p and with r(p, qi) converging to r(p, q). We leave it to the reader to define an appropriate metric on the set of half-lines in ]RN and to prove the lemma. Another geometric tangent structure suggested by the lemma is the following: Definition 2.1.13 For any set S C RN and any point pES we define

to be the the translation to the origin of the union of all lines l(q, q') := {q + t(q' - q) :t E R}

(2.6)

passing through p such that sequences {qi} and {qH of points in Sexist with both qi and q~ converging to p and with l(qi, qD converging to l(q, q'). Again, the reader should define an appropriate metric on the spaces of lines in RN. Example 2.1.14 Let S C ]R2 be the graph of the function

f(x) = { x 2 sin{l/x) ~ ~ i' 0,

o

if x

= o.

Then Tan(S, (0, 0») is the x-axis, but we have

This second notion of a tangent structure has a surprising and not wellknown utility as evidenced by the next result. Theorem 2.1.15 If 0 C ]RN is a non-empty open set, then 0 is a Cl domain if and only if at each point p E ao the following hold: Tan(S,p) = Tan2(S,P), Tan(S,p) is an (N - I)-dimensional plane.

(2.7) (2.8)

As far as we know, Theorem 2.1.15 is due to K. R. Lucas [1]. It is easy to see that, if 0 C ]RN is a C 1 domain, then the conditions (2.7) and (2.8) must hold. The interesting part of the result is the converse, which is a consequence of the following sequence of lemmas.

2.1. THE BUNDLES ASSOCIATED WITH A BOUNDARY

33

Lemma 2.1.18 Suppose S is a closed subset of JRN' and Tan2(S,Po) is a vector subspace of lllN. Then the orthogonal projection onto the translation ofTan2(S,Po) to Po is a one-to-one function on SnU for some neighborhood U ofA». Proof: Assume that there exists a sequence of pairs of distinct points qi and tJi, both converging to Po, such that qi and tij project orthogonally onto the same point of Tan 2 (S,Po). Considering the limit of l(qi, tJi), we are led to the contradiction that there is a line in Tan2(S,Po) that is orthogonal to Tan2(S,Po)· • Lemma 2.1.17 Suppose that S is a closed subset of]RN and that Po is a point of S with Tan(S,Po) Tan2(S,Po).

=

If there is a neighborhood U of Po such that Tan(S,p) is a vector subspace of aN, for all II e S n U, and the dimension of Tan(S, .) is constant on S n U, then Tan(S, .) is a continuous function at Po.

Proof: Assume that there is a sequence of points Pi converging to Po such that Tan(S,pi) does not converge to Tan(S,Po). Then, for large enough i, we can choose qi in S so that the half-line r(pi, qi) makes an angle with Tan(S,Po) exceeding some positive f, independent of i. But then the pairs of points Pi and qi both converge to Po and the lines l(Pi, qi) they determine make angles with Tan(S, Po) exceeding the same f. Passing to a subsequence if necessary, but without changing notation, we may assume that the lines I(p.,qi) converge to a line. That line is in Tan2(S,Po) by definition, but makes angle greater than f with TaiI2(S,Po) by the choice of Pi and qi. That is a contradiction. . • Lemma 2.1.18 Suppose that S is a closed subset of]RN and that Po is a point of S with Tan(S,Po) = Tan2(S,Po). If there is a neighborhood U of Po such that Tan(S,p) is a vector subspace of]RN for all pES n U, and the dimension of Tan(S, .) is constant on S n U, then Po is a relative interior point of the orthogonal projection of Snu onto the translation of Tan2(S,Po) to Po.

Proof: Let II denote the translation of Tan2(S,Po) to Po. Assume that there exists a sequence of points qi e II that converge to Po and are not images of S n U under the orthogonal projection onto II. Suppose r > 0 is such that B(Po, r) C U. For each i, let tJi be the nearest point to qi in II that is in the image of S n B(Po, r) under orthogonal projection onto II. Let pj be a point of S n B(Po, r) that projects to q~.

34

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

=

Hfor infinitely many i we had IPo-1I;1 r, then a subsequence of the p~ would converge to a point p(r) E S that projects orthogonally onto Po and satisfies !Po - p( r) 1 = r. Were this the situation for arbitrarily small choices of r > 0 with i(po, r) C U, we would be able to construct a tangent vector to S at ~ that is orthogonal to II. Plainly, this would be contradictory, and thus we may assume that, for all sufficiently large i, we have !Po - 11;1 < r. For i such that 11; is an interior point of B~,r;), the right circular cylinder over IIn"i(q;, I~ - q;1) does not intersect SnB~, r) in its interior, but does have p~ in its boundary. Since Tan(S,JI;) is of the same dimension as n, there must be a vector in Tan(S,1I;) that is orthogonal to II. Since this happens for infinitely many i, we contradict the continuity of Tan(S, .) proved in Lemma 2.1.17. • Proof of Theorem 2.1.15: The hypotheses of the preceding three lemmas are satisfied. Thus, near each point of S, we can use the inverse of orthogonal projection onto the tangent plane to define a function onto S. The definition of Tan(S, .) guarantees that this fun~ion is differentiable, and we know that the derivative is continuous because of the continuity of the tangent planes proved in Lemma 2.1.17. •

2.2

The Second Fundamental Form and Curvature

Recall that the Gauss map for the boundary of a domain function p is given by Vp(x) 9(x) = IVp(x)l"

n with defining

Since 9(x) is a unit vector normal to an at x, then it must be independent of the choice of defining function. Thus we have the the next result. Proposition 2.2.1 If P is another defining function for n and if 9 is the Gauss map defined using p, then 9(x) = 9(x)

for x E

an.

(2.9)

This is readily verified using Proposition 1.2.3. Assume that p is at least C2. Notice that, for any vector v, the directional derivative of 191 2 == 1 in the direction of v is zero, so we have

9· (v,D9) == O. Accordingly, if x is a point of tangent to an.

(2.10)

an and v is any vector, then (v, D9(x)}

is

2.2. THE SECOND FUNDAMENTAL FORM AND CURVATURE

35

Definition 2.2.2 Suppose n is C k , k ~ 2, and x E an. The Weingarten map W., : Tan(an,x) -t Tan(an,x) is defined by setting W.,(v) equal to the directional derivative of the Gauss map in the direction of v, that is

W.,(v) = (v,DQ(x»,

(2.11)

where v is tangent to an at x. Remark 2.2.3 Since it has been noted that (v, nQ(x» is always tangent to an for x E an, even when v is not a tangent vector, one might hope or assume that if v is normal to an, then (v, nQ(x» would vanish. Such is not the case. Perhaps the simplest example to illustrate this is to take p(x,y) = y - xy

to be the defining function for {(x, y) : y < O} in some small neighborhood of the origin. We leave it to the reader to verify by direct calculation that, with v = (

~

) the upward pointing unit vector, (v, nQ(O, O)} = (

-~

) .

Proposition 2.2.4 Suppose n is Ck, k ~ 2. If P is another defining function for n and if W is the Weingarten map defined using p, then W.,(v)

whenever v is tangent to an at x

== W.,(v),

(2.12)

E~~.

The proposition is most simply proved by a computation using coordinates. While such computations can be tedious, they are invaluable for understanding examples. For this purpose, we introduce some notation. Definition 2.2.5

(i) The standard basis vectors in]RN will be denoted by

(ii) For a defining function p, we will write P,i P,ij

ap = -, i E {1,2, ... ,N} aXi 2p = -aa a ' i,j E {1,2, ... ,N}. Xi

Xj

Lemma 2.2.6 Suppose that p is C k , k ~ 2.

(2.13) (2.14)

36

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

(i) For any pair 0/ vectors v have

= 2:;:1 vie;

and w

= E!1 Wiei

in RN, we

and

(ii) If, in addition, w is a tangent vector, then (2.17) .

Proof: In terms of components we can write N

g(x) =

L Giei, i=1

with

(2.18) We have (v,Dg(x))

=

LNaG· ax~Vjei i,j=1

J

with

~~~ = P,ij (t(p,d) 1

k=l

-1/2 _

P,i

(t k=l

P,jkP,k)

(t k=1

CP ,k)2) -3/2, (2.19)

2.2. THE SECOND FUNDAMENTAL FORM AND CURVATURE

37

so (2.15) follows. Equation (2.16) follows readily from (2.15), and then (2.17) follows from (2.16). • Proof of Proposition 2.2.4: By Proposition 1.2.3, there exists a positive function h of class CIe-l such that

=

P P,i

=

hp ah - haz.P+ P,i

P,i;

=

- ah ah h ,iiP+ -8 p,; + -a P,i Xi

X;

+ h-P,i;·

Now the last equation only makes sense if h is twice differentiable, i.e. if k ~ 3, but even in case k = 2 we have

= =

P,i P,i;

hp,i ·ah ah -a p,; + -a P,i Xi

X;

+ h-P,i;

for points in an, in particular, at x. Since v is tangent to an at X, we have N

N

L P,iVi i=1

=L

P,iVi =

o.

i=1

Thus N

LP,ijVj

;=1

N

(ah _

ah _

_)

= ;=1 L a. P'; + 8x.J P,i + hp,ij

Vj

X.....

N

=

ah

L ap,i i=1 Xj

V;

+

N

E hp,ijv; i,;=1

(2.20)

and N

E

P,ikP,k V;

=

(ah _ ah _ _)_ L {}":p,k + ap,j + hp,ik hP,kVi i,k=1 Xk N

XJ

;,k=1

N

=

N

~ ah h-2 " h 2P,jkP,k - - V;. ~ {}": P,k V; + 'L.J i,Ie=1

XJ

(2.21)

;,k=1

Substituting (2.20) and (2.21) into (2.15) and doing a bit of algebraic manipulation, we find that

38

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

("f. p.

iei)

i=1

=

(t

PJleh.Vi)

i."=1

1e=1

.

•

(V, Dg(z».

Corollary 2.2.7 Let Gi, i map.

= 1,2, .. . ,N, be the components of the Gauss

(i) For any pair of vectors v = E!1 Vil!j and w = have

w·(v,Dg(z»

E!1 Wiei in JtN, we

NaG·

=L

{}

i.j=1

(ii) For j

(E(p.Ie)2) -3/2

~WiVi

(2.22)

XJ

= 1,2, ... ,N, we have (2.23)

Proof: Part (i) follows from the proof of Lemma 2.2.6. Part (ii) then follows from (2.10) with v = ej. • Definition 2.2.8 Suppose that n is C", k ~ 2, and that z E an. The second fundamental. form is the bilinear form n on Tan(an, x) given by

n(v,w)

=w· W",(v)

(2.24)

where v and w are tangent to an at x. Proposition 2.2.9 The second fundamental form is symmetric. Proof: This follows from the symmetry of the second derivative and (2.17) .

•

The second fundamental form is the quadratic form associated with the Weingarten map. Since the second fundamental form is symmetric, we know from linear algebra that the Weingarten map will have only real eigenvalues and that there will exist a set of N - 1 orthogonal eigenvectors. Also recall from linear algebra that a linear map from an N - 1 dimensional vector space to itself has associated with it a set of N - 1 invariants. The two most familiar and most commonly used of these invariants are the trace and the determinant. One definition of the other invariants is given next.

2.2. THE SECOND FUNDAMENTAL FORM AND CURVATURE

39

Definition 2.2.101/ a linear map from an M dimensional real vector space to' itself is represented by the M x M matrix A, then the trace of order K of A is denoted by trK(A) and equals the sum 0/ all the K x K determinants that can be formed by intersecting any K rows 0/ A with the same K columns. That is

trK(A) =

L

t1i1il

ail i,

Gil iK

Gi,il

ai,i2

Gi,iK

aiKit

aiKi,

GiKiK

where the sum extends over all choices 0/1 :5 i~

< i2 < ... < iK :5 M.

Note'that the usual trace of A equals the trace of order 1 of A and the determinant of A equals the trace of order M of A if A is M x M. Definition 2.2.11 For x E an, (i) the (N - 1) eigenvalues o/W", are called the principal curvatures o/an,

(ii) the unit eigenvectors 0/ W", are called the' directions of curvature or principal directions o/an, (iii) the determinant 0/ W", is called the Gaussian curvature or total curvature o/an and will be denoted by K(x), (iv) the number N~l times the trace o/W", is called the mean curvature o/an at x and will be denoted by H(x),

(v) the number (N_l)1(N_2) times the trace of order 2 0/ W", is called the scalar curvature o/an at x and will be denoted by Sex). By (2.2.7), we know that the second fundamental form is the restriction to the tangent plane of the quadratic form associated with the matrix ~, given in (2.19). Additionally, we know that the range of that matrix is the tangent plane. The following lemma from linear algebra will allow us to use these facts to compute the mean curvature, the Gaussian curvature, and the sum of the squares of the principal curvatures.

Lemma 2.2.12 Suppose A is an N x N matrix 0/ rank N - 1 with range denote the linear map from II to II that is the restriction to II 0/

n. Let A

the linear map determined by A.

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

40

(i) For K

= 1,2, ... , N -1, the trace 0/ order K 0/ A

order K

0/ A,

(ii) The trace

equals the trace

0/

that is,

0/ A2

equals the trace tr[A2]

0/ A2,

that is,

= tr[A2 ].

Proof: Since the left-hand and right-hand sides of both equations are independent of choice of basis, we may use any bases that we find convenient. Choose an arbitrary set of N - 1 independent vectors in II and extend it to a basis for JllN by adding a null vectOr·of A as the Nth basis vector. With such a basis, A is represented by an N x N matrix with Nth row o and Nth column 0, while A is represented by the (N - 1) x (N - 1) matrix obtained by omitting the Nth row and Nth column of the matrix representing A. Part (i) is now evident. Part (ii) follows by considering the matrices representing A2 and A2. • Definition 2.2.13 For a C2 function / defined on a domain .in JllN, we define the Laplacian of /, denoted 6./, by .

B2/ L-a N

6./=

i=l

2· Xi

Theorem 2.2.14 Let 0 be a domain with defining function p E Cle k ~ 2. Let the tangential differentiation operators Oi be as in (2.9). For x E an; the mean cUnJature at x is given by N

H(x)

=

(N - 1)-1

L aG

i

i=l aXi

N

=

(N - 1)-1

L OiGi i=l

=

(N

-W'IVpl-' (IVPI'

L'.p(.)

-;t

p,;(')PJ(') P,;;('»)

where the tangential differentiation operators Oi are as in (2.9). Proof: Fix a point Xo E ao. Applying the preceding lemma to the Weingarten map, we see that N

(N -1)H(x)

aG.

= L~. i=l

X,

2.2. THE SECOND FUNDAMENTAL FORM AND CURVATURE

41

fl:- by the expression given in (2.19),

Multiplying by IVp(x)j3, replacing and simplifying, we obtain the result.

•

Corollary 2.2.15 Let 0 be a domain with defining function pEe" k ~ 2.

1/ IVpl=1

0/ a point x E 80,

holds in a IpGtial neighborhood

then

H(x) = N~l b. p(x). Proof. We have

N

2:P~i == 1 ;=1

in a neighborhood of x. Differentiating with respect to x;, we find that N

2:p,;p;; = 0 ;=1

holds for each j. Multiplying QY p,; and summing over j, we find that N

2: P,;P,;P,;; = o. ;,;=1

The result then follows from the last line in Theorem 2.2.14.

•

Lemma 2.2.16 Let the tangential differentiation operators 6; be as in (2.3). For any 1 ~ i,j ~ N, we have

6;G;

= J;G;.

(2.25)

Proof: Using (2.19), we see that

6;G;

a

a 2: G,,-G; k=l aXk N

=

-G; - G;

=

IVpl- 1 P,;; - IVpl- 3 P,;

ax;

N

2: P,kP,;" k=l

N

-

IVpl- 2 p,;

2: P," (IVpl-

1

N P,;" -IVpl- 3 P,; ~:>,'P,"l)

"=1

1=1

N

=

IVpl- 1 P,;; .,.. IV'pl- 3 P,;

3

"=1 N

+

IVpl- 5 P,;P,i

2: p,,,p,,,tP,t· ",t=l

N

2: P,,,P,;" -IV' pr p,; E P,kP,;" 10=1

42

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARlES

Since the last expression is clearly symmetric in i and;, the result follows .

•

c"

Theorem 2.2.17 Let n be a domain with defining function p e Ie ~ 2. Let the tangential differentiation opemtors 6. be as in (B.S). For x e ao, the sum of the squares 0/ the principal cUnJatures is given by N

~ .aG. aGj L..J3---i=1

aXj ax,

N

"" 1£ G L..J ". j

= ',j=1

12 .

fI!:.

Proof: Let G be the matrix with entries By Lemma 2.2.12, we know that the sum of the squares of the principal curvatures equals tr[G2]. But the (i, Ie)th entry in G 2 is

so

as desired. . To see that L~=1 16i G j l2 also equals the sum of the squares of the principal curvatures, we use the Ilreceding lemma to compute N

E 16.G;1 = 2

',j=1

"j=1

= =

EN .. 1 1,3=

EN

aGj aGo _ aGj GjGt aGo ax. aXj ',3, . "I = 1 ax. aXI

Now, by part (li) of Corollary 2.2.7, we see that N

L ".[

'J,

=1

8Gj GjGt aG. ax. aXt

=

=

2.2. THE SECOND FUNDAMENTAL FORM AND CURVATURE

43

EN GIeGl (N E aG)(N EO; 00) a;

le,l=1

i=l

Gir Xl

0,

=

Xle

;=1

.

•

proving the result.

The following theorem is an immediate consequence of Lemma 2.2.12. It makes use of the trace of order N - 1 which is a matrix invariant defined in Definition 2.2.10. Theorem 2.2.18 Let {} be a domain with defining function P E Cle k ~ 2. Let X be a point in a{}.

(i) The Gaussian cUnJature at x is given by K(x) = trN-l

[aG.] ax; .

(ii) The scalar cUnJature at x is given by Sex)

= (N

deN 2)tr2

[~~;].

Example 2.2.19 Let {} be the region with defining function

P(Xl,X2, ... ,XN) = (tx~) - R2, 0=1

so that {} is the interior of the ball of radius R centered at the origin. One computes that P,i

=

2x;, 21S.;,

P,,; =

where lSi; is the Kronecker delta. It follows that at any point x E

a{}

8G. IS.; XiX; 8x; = If - R3 holds. It is then routine to compute that

EN 8G. = N - 1

EN aG. aG; = N -

1,

ax; ax. R2 so at every point of a{} the mean curvature is II and the sum of the squares . 1

1=

ax.

and

R

.. 1 ' 3= t

of the principal curvatures is ~. The computation of the Gaussian curvature and the scalar curvature is a bit less straightforward. One can use the fact that

44

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

for y

= (YlJl/2, ... ,YM) E R.M, the M

x M matrix

IMxM _yyt has the eigenvalues 1 and 1 - Iv1 2 ; with the multiplicity of the eigenvalue 1 being M -1, where yyt is the matrix with (i,j)th entry YiY;, to verify that the Gaussian curvature of an has the expected value of and that the scalar curvature is at every point.

b,

RA-i

Plane Curves Consider a region n in the plane. n of class C k , k ~ 2, satisfying

Suppose that P is a defining function for

IVpl·= 1

in a neighborhood of length of an .

an, and suppose res)

is a parametrization byarc-

Definition 2.2.20 The curvature of an is the non-negative function defined by setting

/'i,=I~~I·

/'i,

(2.26)

Writing the components of r as xes) and yes), we can differentiate the equation p[r(s)] = 0 with respect to s to find

o = o =

p,IX' + P,2Y', P,ll X,2 + 2P,l2 X' Y,

(2.2~)

+ P,22 y + P,IX + P,2Y

Likewise, we differentiate the equation X,2

,2

"

"

.

(2.28)

+ y,2 = 1 to obtain

0= x'x" + y'y".

(2.29)

Now, (2.27) and (2.29) can be thought of as a pair of linear equations simultaneously satisfied by (x', V'), so the determinant of the coefficients of that system vanishes, that is,

(2.30) Equation (2.30) tells us that (x", y") is orthogonal to (P,2, P,l) which in turn is orthogonal to the unit vector (P,l, P,2). Since we are in a tw()odimensional space, we must have l(p,lJP,2)· (x", y")1 = I(x" ,y")1 or

Ip,lx" + p,w"l

= J(X")2 + (y")2.

(2.31)

2.3. SURFACES WITH CONSTANT MEAN CURVATURE

45

Similarly, we can differentiate the equation P;l + P;2 = 1 with respect to the first and second variables to find

o = o =

P.IP.ll + P.2P.12, P.IP.12 + P.2P.22·

(2.32) (2.33)

Equations (2.27) and (2.32) together imply

o= P.llY' -

P.12X',

'(2.34)

P.22X'.

(2.35)

while (2.27) and (2.33) imply

o= P.12Y' -

Multiplying (2.34) by Y' and (2.35) by x' and adding the results to (2.28), we obtain (2.36) 0= P.ll + P.22 + P.IX" + P.2Y"' and from (2.31) and Corollary 2.2.15 it follows that the mean curvature satisfies the equation IHI = ..j(XIl)2 + (yll)2. (2.37) We can summarize .the above calculations in the next theorem, which also embodies the part of the Frenet formulas appropriate to curves in the plane. Of course, there is a more general set of Frenet formulas applicable to curves in space1 (see for example Struik [1]), but that will not be presented here. Theorem 2.2.21 For'a C k , k ~ 2, domain n c lR? with by arc-length by res), the following equations hold K,

= IHI,

an

parametrized (2.38)

(2.39)

2.3

Surfaces with Constant Mean Curvature

From Theorem 2.2.14 of the previous section, we know that if P is a function satisfying "V P -lOon the surface S = {x : p( x ) = O}, then the mean curvature of S is given by the differential expression (2.40) 1 More general Frenet formulas for curves in spaces of arbitrary dimension can be found in Shilov [1] or in Green [1].

46

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

evaluated on S. It is interesting to consider what happens when we reverse this· process by first specifying the mean curvature and then seeing what the consequences are for the surface. The most natural specification for the mean curvature is to set it equal to a constant. Thus we will be considering the equation N

IVpI 2 l:J. P-

L P,lPJP,li = C{N -l)lVpI3,

(2.41)

l,i=1

where C is a constant. We saw in the previous section that a sphere of radius I/C is a surface with mean curvature C at every point of the surface. Thus the ball provides an example of a domain with positive constant mean curvature on the boundary. In Example 2.3.6 we will see that the right circular cylinder is another example of a surface with constant mean curvature. The serious study of constant mean curvature surfaces dates back to the 1841 paper of Delaunay [1], in which he studied the rotationally symmetric constant mean curvature surfaces in JR3, now called Delaunay surfaces. Clearly, the sphere and the cylinder are examples of rotationally symmetric constant mean curvature surfaces, but we will use the term "Delaunay surface" to mean those constant mean curvature surfaces other than the sphere and the cylinder; this choice of terminology is a matter of taste. Definition 2.3.1 By a surface in R" we mean a set S C"JR3 such that for each point pES there is an open set U C JR3 such that SOn U with the relative topology is homeomorphic to an open subset of 1112. The topological type of a surface in R' that bounds a domain is characterized by two numbers: The genus and the number of ends. A useful heuristic. definition of the genus is that a surface is of genus 9 if it is homeomorphic "to a sphere with 9 handles attached. A more precise definition is as follows: Definition 2.3.2 If a surface is homeomorphic to a sphere, then its genus is 0; if a surface is homeomorphic to a torus, then its genus is 1; and if a surface is homeomorphic to a connected sum of 9 ~ 2 tori, then its genus is g. Remark 2.3.3 It takes some effort to make precise the notion of "connected sum" used in Definition 2.3.2; the interested reader may consult Bloch [1], Chapter 2, fot example. The Classification Theorem for Surfaces tells us that every oriented, compact, connected 2-dimensional surface has a genus as in Definition 2.3.2. Definition 2.3.4 Let S be a surface. If there exist a compact, connected 2-dimensional surface S' and m distinct points Pl,]J2, •.• ,Pm E S' such that S is homeomorphic to "

s' \ {p,.,P2, ... ,Pm}, then S is said have mends.

2.3. SURFACES WITH CONSTANT MEAN CURVATURE

II compact 1 end 2 ends 3 or more ends

genus 0 . I genus> 1 Round spheres None None None Cylinders None Delaunay surfaces Kapouleas surfaces

47

I

Imbedded Surfaces with Constant Positive Mean Curvature Table 2.1: Existence of Surfaces Recently N. Kapouleas [1], [2], [3] has 0, then there exists a C le junction f, defined on a ball in RN-l centered at ~,P2, ... ,PN-d such that

(x,f(x») E an for all x E B, j(PloP2, ... ,PN-d

=PN

(2.42) (2.43)

and

Proof: It is immediate from the Inverse Function Theorem that locally there exists a C le function, f, satisfying (2.45) 2The author's name is also transliterated as Alexandrov.

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

48

and satisfying (2.43). Of course (2.45) is equivalent to (2.42). Now we set U(Xl,X2, •.• ,XN)

= XN -

I(X~,X2, ••• ,XN-l).

The level set U = 0 gives 00 near P, and every other level set of U is simply a translate of 8n in the x N direction, with the same normal direction as the original surface. Thus the mean curvature operator applied to U will produce the value of the mean curvature at the corresponding point on an. That is, we have

Noting that

au 8xj 8u 8XN 82u 8xj 8xIc 82u 8xj 8xN

=

al -aXj

=

1,

=

al - 8xj 8x/c

=

0

for 1 ~ i

for 1 ~ i

~

N - 1,

for 1 ~ i, k ~ N - 1, ~

N,

we observe that (2.46) implies (2.45).

•

Example 2.3.6 The right circular cylinder of radius R with axis in the Xl-direction has the defining function . . -P(X1,X2,X3) = x~ + x~ - R2.

The condition I!; > 0 is satisfied on the top of the cylinder, that is, for X3 > o. The appropriate function I(X1,X2) is I(X1,X2)

= VR2 -x~.

Computing with (2.44), we find the mean curvature of the surface is 2~. Now let us apply Lemma 2.3.5 to a surface of constant mean curvature in

JR3 that is rotationally symmetric about an axis. We will write x = Xl,

=

Y = X2, and z X3. Suppose that the surface is symmetric about the x-axis and that the surface has constant mean curvature C ~ 0 when considered as the surface bounding the region containing the x-axis. If rex) is the function describing the radius of the cross section of the surface perpendicular to the x-axis, then we have

2.3. SURFACES WITH CONSTANT MEAN CURVATURE

49

We compute

oz Oz = OZ = Oy

r r' (r2 - y2)-1/2, _y (r2 _ y2)-1/2,

{}2z

IJz2

=

r r" (r2 - y2)-1/2 - y2 (r')2 (r2 _ y2)-3/2,

{}2z 0z0y

=

yrr'(r2 _ y 2)-3/2,

{}2z

=

_r2 (r2 ...:. y2)-3/2.

Oy2

Substituting these equations in (2.44), we are led to the equation r"

= 1 + (r')2 r

_ 2 C [1 + (r')2]3/2,

(2.47)

which must be satisfied by the radius of a surface of constant mean curvature symmetric about an axis. .We have proved the following proposition. Proposition 2.3.7 If rex) >.0 is a C", k ~ 2, function such that the . boundary surface of the region S

= {(x, y, z) : y2 + Z2 < r(x)2}

has constant mean cUnJature C

~

0, then rex) satisfies the equation (2.47).

While the existence and uniqueness theorem for ordinary differential equations applies to (2.47) for all values of C, it is unfortunately the case that only when C = 0 can a solution be found relatively easily. We summarize the facts for the case C = 0 in the following corollary, which the reader can verify.

>0

Corollary 2.3.8 Ifr(x) face of the region

is a C2 function such that the boundary sur-

has zero mean cUnlature, then rex) is given by rex)

=

Q

cosh(-yx) + f3 sinh(-yx),

where If, additionally,

ro

= r(xo)

and rl

= r'(xo),

ou

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

then.

Jl+rf ro ro cosh('Yzo) - rl sinh(-yzo), a 'Y rl cosh(-yio) ro sinh(-yzo). (3 'Y Remark 2.3.9 A surface having mean curvature zero is called a minimal surface because any smooth surface that locally minimizes area must have mean curvature zero. There are many interesting results and problems involving minimal surfaces, but we will not pursue them in this book. We refer the reader to Osserman [2] and Nitsche [1]. 'Y

o

= = =

0-

Delaunay Surfaces

o

For our further consideration of rotationally symmetric constant mean curVature surfaces, we will assume that C > 0 and that r' in (2.47) is not identically zero. A first integral of (2.47) can be found by multiplying both sides of (2.47) by r' and cQmputing

2Crr'

so

=

r' [1 + (r')2]-1/2 - rr' r:' [1 + (r,)2r 3/ 2

=

:/x {ro[l + (r')2]-1/2} , VI + (r'}2 =

r

.

Cr2+a While one might anticipate eventually finding a solution for (2.48), it would seem unlikely that the solution would be more revealing than the equation. Delaunay's amazing discovery is that the solution can be described nicely. Pursuing this assertion requires that we treat an apparently unrelated topic. A roulette is a curve generated by a point on a rolling curve (the curve mayor may not be closed). The best known example of a roulette is the cycloid which is the curve generated by a point on the circumference of a circle as the circle rolls without slipping along a straight line. The epicycloid is the curve generated by a point in the interior of a circular disc as the disc rolls without slipping along a straight line. The hypocycloid is similarly generated by a point exterior to the circle as the circle rolls without slipping along a straight line. We will also consider the epicycloid and the hypocycloid to be roulettes of a circle. To consider a general roulette, let r be a plane curve and let q be the point which is to generate the roulette. Thus, the curve r is assumed to be in contact with a line L, and r is to be acted on by rigid motions, say parametrized by 8, so that r and L remain in contact with the point of contact between r and L, as a function of 8, giving an arc-length parametrization of both r and L. The rigid motions constitute the "rolling" and the

2.3. SURFACES WITH CONSTANT MEAN CURVATURE

51

requirement that the contact point gives an arc-length parametrization of both the curve and the line is what is meant by rolling "without slipping." The roulette is the path of q under the same rigid motions associated with rolling without slipping. The point q need not be a point on 1'. Assume that p(s) is the vector from q to a point on the curve giving an arc-length parametrization of 1'. The easiest approach in deriving the equation of the roulette is to consider a line, L, rolling without slipping along the curve I' and to then give the coordinates of q relative to the line. Assume the initial point of contact between-I' and L is p(O). We also associate with L a right-handed cartesian coordinate system with coordinates z and 11 as follows: The coordinate system associated with L will have L as its z-axis, the point z = 0 will be at the initial point of contact, p(O), and the positive x-direction will be the unit tangent direction at the initial point of contact, that is, yeO). The y-axis is then chosen to give a right-handed coordinate system. As the line L rolls along r without slipping, the point of contact between the line and the curve at pes) will have z-coordinate s. The positive zdirection will agree with y(s), the unit tangent vector to I' at s, denoted T(s). The vector from the point of contact of I' and L to the point q is -pes), so the x-coordinate of q is given by xes) = s - pes) ·T(s).

Recall from Definition 2.2.20 that the curvature", of I' is

The behavior of the y-coordinate of .q..is simplified by assuming that the curvature of I' does not vanish at any point and that the unit tangent vector T and the unit normal vector N given by N=

elI' ds

",-1_,

form a right-handed system. In that case, we have yes) = -pes) ·N(s).

We have proved the following proposition. Proposition 2.3.10 IjtheC", k ~ 2, cunJe I' has non-vanishing cUnJature and the arc-length parametrization pes) is such that the unit tangent vector T(s) and unit normal vector N(s) form a right-handed system, then the roulette of the point q at the origin is given parametrically bll xes)

=

s - pes) ·T(s),

(2.49)

yes)

=

-p(s)· N(s).

(2.50)

52

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

Figure 2.1: The Cycloid Example 2.3.11 Let a circle of radius r be parametrized by ( ) _ ( -a + r cos(s/r) ) r sin{s/r) ,

p s -

where a > O. The center of this circle is at the point with coordinates (-a,O), so the point q that will generate the roulette is a units to the right of the center of the circle. We compute T(s) - '(s) _ ( - sin{s/r) ) - p cos(s/r) ,

N(s)

= ( --sm{s/r) ~s(s/r)

) .

So the parametriz.ation of the roulette is x(s) = s - a sin(s/r), y(s) = r - a cos(s/r).

When 0 < a < r this roulette is an epicyc1oid, when a = r it is a cycloid (see Figure 2.1), and when r < a it is a hypocycloid. In practice, it is usually not particularly advantageous to eliminate the parameter s from the pair of equations (2.49) and (2.50) for x(s) and y(s'), the previous example being a case in point. Notice that, since we are working in the plane, the direction of the yaxis is determined by the direction of the x-axis. By making a convexity assumption, we can obtain a form for the parametrization of a roulette that avoids the requirement that the curve r be parametrized by arc-length and that even applies if r is only C1 • One should visualize the curve r and the region it encloses rolling along a straight line. We consider the curve traced out by a chosen interior point q, and for convenience of notation we suppose that that interior point is initially at the origin.

Proposition 2.3.12 II the Cle, k ~ 1, curve r encloses a convex region with the origin in its interior, and is parametrized in the positively oriented direction by p{t), then the roulette of the point q at the origin is given parametrically by x(t) y(t)

= =

s(t) -lp'(t)r1p(t)· p'(t),

(2.51)

v'lp(t)jZ -1P'(t)1 2(P(t)· P'(t»2,

(2.52)

2.3. SURFACES WITH CONSTANT MEAN CURVATURE

53

where set) is the arc-length from p(O) to pet).

Proof: We know that the unit tangent vector is given by 1P'(t)I-IP'(t), so the formula for z(t) is immediate. We also knOw that.

where N is such that T and N form a right-handed basis. Since the origin is interior to the region enclosed by r, we know that p N must be negative, and the formula for yet) follows. • 0

Looking at Propositions 2.3.10 and 2.3.12, we may think that the roulette has lost one order of differentiablity compared to the original curve, but that is an illusion. In fact, we will see that a roulette is as smooth as the given parametrization of r : Proposition 2.3.13 If the cUnJe r htJs non· vanishing curoature, encloses a convex region with the origin in its interior, and is parametrized by arclength in the positively oriented direction by the C,., k ~ 3, function pes), then the roulette of the point q at the origin satisfies dx ds dy ds

rPz ds 2 rPy ds 2

=

-K.p·N> 0,

(2.53)

=

K.poT,

(2.54)

=

2 dK. K. p·T- ds p·N,

(2.55)

=

dK. 2 K. + dsP-·T + K. p·N.

(2.56)

Moreover, the roulette is the graph over the x·axis of a positive C,. function satisfying the differential equation dy y dx =F(x), where F(z) is a C,.-l function.

Proof: Equations (2.53) - (2.56) follow from (2.49) and (2.50) and the Frenet formula (see Theorem 2.2.21) dN

Ts = -"T. From (2.53) and (2.54) it follows that dy = dx

pes) oT(s) p(s)oN(s)'

54

CHAPTER 2. DOMAINS WITH SMOOTH BOUNDARIES

so from (2.50) we have

dy Ydx =p(s)·T(s).

(2.57)

By (2.53) we can apply the Inverse Function Theorem to (2.49) to express s as a C k - 1 function of x and consequently p(s)· T(s) is a C k - 1 function of x, which we denote by F . ' • From the proof of the theorem we obtain the following useful corollary.

Corollary 2.3.14 -Under the conditions of Theorem 2.9.19 the function y(x) describing the roulette satisfies

YV1

+(:Y

=IPI·

Proof: We use (2.50) and (2.57) to compute

Ipl2 = =

(p. N)2 y2+

+ (p. T)2

(y:r

•

Exrunple 2.3.15 We consider the roulette of an ellipse about one of its foci (see Figure 2.2). Let an ellipse with major axis of length 2a and minor axis of length 2b be parametrized by P

where a 2

( fJ)

= ( -c + ~ cos fJ

) bsm9'

= I? + ~, thus the foci are (-2c, 0) and p

'(fJ)

=(

(0,0). We compute

-a sinfJ ) . b cos 9

A little manipulation then gives us

Ipi Ip/l p' p'

=

a-ccos9,

(2.58)

[a 2 - C cos2 (9)j1/2,

(2.59)

c sin 9 [a - c cosfJ].

(2.60)

Using (2.52), we find that

y=b

a-ccosfJ a+ccosfJ'

or, if we solve for cos fJ, alJ2-y2

cosfJ= - - - . c b2 + y2

(2.61)

2.3. SURFACES WITH CONSTANT MEAN CURVATURE

55

Figure 2.2: Roulette of the Focus of an Ellipse Since (2.51) will produce an equation for x in terms of cos 8 and sin 8, we could use (2.61) to eliminate 8 and obtain a solution. Unfortunately, the solution obtained involves an elliptic integral. An alternative approach is to use Corollary 2.3.14 to write (2.62) so we can. eliminate cosO from equations (2.61) and (2.62) and see that the roulette of an ellipse generated by a focus of the eIDpse satisfies the differential equation

J (d 1+

y )2 dx

2ay y = y2+b2 = ..Ly2+~· 2a 2a

(2.63)

Notice that the differential equation satisfied by the roulette of an ellipse (2.63) is the same as the differential equation (2.48) satisfied by the function r(x) describing a rotationally symmetriC-surface with constant mean curvaThus we have obtained an aesthetically pleasing description ofthe ture solution of (2.48). In general, Delaunay discovered that a roulette formed by the focus of any conic section gives the longitudinal cross-section of a rotationally symmetric surface of constant mean curvature. The interested reader should also consult the article Eells [1].

fa.

Chapter 3

Measures 3.1

The Caratheodory Construction

Let :F be a collection of subsets of IIlN. Let (1'::F -+ {x E Ill: x ~ OJ.

We refer to the function (1' as a gauge. We need to make the following two assumptions concerning :F and (1' : For 6 > 0, each E C IIlN has a covering by a set of at most (3 1) countably many members of :F having diameter less than 6. . 0=

m( sup{(1'($) : 8 E :F, _~iam(8) < 6}).

Caratheodory's idea for constructing a measure from :F and follows:

(3.2) (1

is as

Definition 3.1.1 (Caratheodory [I» (i) Let E C JRN be any set. Let 6 > o. Call 0 c :F a 6-admissible cover of E if 0 has at most countably many elements, each element of 0 has diameter less than 0, and finally

(3.3) Let.A.J

=A.J(E) denote the collection of allo-admissible covers of E.

(ii) Set 1l6(E) = inf

E (1'(8).

(3.4)

OEA'SEO

We call1l6 the size 6 approximating measure associated with :F and (1.

57

CHAPTER 3. MEASURES

58 (iii) Set

'H.(E) = lim 'H.6(E). 6-+0+

(3.5)

We call1l the result of Caratheodory's construction from :F andu. By (3.1), 6-admissible covers will exist for any 6 > 0 and any E C aN. Also, notice that, by definition, if 0 < 61 < 62 , then 1l61{E) ~ 1l62(E), so the limit on the right-hand side of (3.5) exists. The condition (3.2) is included to insure that 'H.6(0) 'H.(0) 0, though some may feel this is redundant since the empty cover covers the empty set, and the empty sum is zero. We leave it to the reader to verify the following easy result.

=

=

Lemma 3.1.2

(i) For any 6 > 0, 'H.6 is a measure, and i/ all elements o/:F are Borel sets, then every E C aN is contained in a Borel set B with 1l6(B) = 1l,,(E). .

(li) 1l is a measure, and i/ all elements 0/ :F are Borel sets, then 1l is a Borel regular measure. We turn immediately to the applications of CaratModory's construction that are of greatest importance. Notation 3.1.3 We will use T" to denote the volume 0/ the unit ball in IR". It is known (see Proposition A.2.6) that 7r k / 2

Tk

= 2k~{k/2r

Let :F be the collection of open Euclidean balls in IRN. For B = B(x, r) E :F and k = 1,2, ... , let Uk (B) = T krk. Since T k is the volume of the unit ball in IRk, we see that uN{B) is actually the Euclidean volume of B. One advantage of this last formula is that the right-hand side is defined for any positive real number a (instead of k) and allows us to define T Q for positive real a's. We also define U for any positive real number a by Q

7rQ/2rQ

ucr(B) = 2 ar(a/2)'

(3.6)

The result of Carathoodory's construction from this :F and this Ucr will be denoted by SQ. This measure is the a-dimensional spherical measure on IRN • Let us consider spherical measure by way of several examples. Example 3.1.4 Let E

c a be a segment of Euclidean length t. Then

Scr{E) =

{

+00 if a < 1 ~

if a = 1 if a>1.

3.1. THE CARATHEODORY CONSTRUCTION

59

Figure 3.1: Covering a Segment by Euclidean Balls To verify this assertion, we first make the geometrically obvious (and not difficult to check rigorously [Hurewicz and Wallman, 1]) statement that the most efficient way to cover E with Euclidean balls of radius less than 6 is to lay them side by side with arbitrarily small overlap; see Figure 3.1. Taking the balls to be of radius about 6, we see that there are about £/(26) of these balls. Now if Q < 1 then we see that the sum in (3.4) is about (3.7)

As 6 -+ 0+, we find that this expression converges to +00. IT Q > 1 then the calculation is the same, but now the expression in (3.7) converges to O. Finally, in case Q = 1 then (3.7) converges to Tl . £/2 = t. Example 3.1.5 . Let us investigate how the last example changes if the segment E is now considered as a subset of JR.N . The beauty of this theory is that the picture (Figure 3.1) and the calculations from the last example are exactly the same. The result is __ SO(E)

={

+00 if.a < t

o

1

if a = 1 if a>1.

Example 3.1.6 IT E C litH is any set and if a> N then So (E) = o. To see this, first assume that E is a closed cube of side length £. For any positive integer k, we can subdivide the closed cube E into kH closed subcubes of side length t/k that intersect only on their boundaries. Each subcube can be covered by a closed ball of diameter .JNt/k with center at the center of the subcube; see Figure 3.2. So for 6 slightly larger than .j!ij£/k the sum in (3.4) becomes (3.8) Since Q > N, it is easy to see that this expression tends to 0 as k -+ 00, thus as 6 -+ 0+. Since So is subadditive, the full result follows.

CHAPTER 3. MEASURES

60

~

-

..........

...--....~

(

L(

) (

(

r--l

0

)

r.-.

]

( ( (

) )

)

(

~

)

'--

-

----

)

Figure 3.2: Covering a Cube by Euclidean Balls Example 3.1.7 Let E C RN be a compact set with interior. IT a < N then sa (E) = +00. To see this, first assume that E is the closure of a cube. We proceed as in the last example, subdividing the cube into subcubes of side length ilk. For 8 less than or equal to ilk, the side length of the subcubes, each ball can contain the center of no more than one subcube, so the sum in (3.4) is at least (3.9)

Since a < N, the expression (3.9) tends to +00 as 8 ~ 0+. Finally, if E is any compact set with interior then E certainly contains the closure of a cube. Since sa is subadditive, we are done with the general case. Example 3.1.8 IT Q C ]RN is the cube Q

= {(XI, •.. ,XN) E RN

then we have sa(Q)

={

:0 ~

Xj

~ I}

a N.

+00 if 1

o

The first and last cases are covered by the preceding two examples. Thus we need only treat the case a = N. In order to keep the length of this calculation under control, we shall invoke some facts from standard measure theory (see Federer [4J for details). Recall from Definition 1.3.20 that eN denotes the usual Lebesgue measure in IRRN. Now fix a 8 > O. Also fix a small T > O. By the Vitali covering lemma (see Section 3.5), we see that there are finitely many pairwise disjoint open balls Bj, all of radius less than 8, that lie in Q and are such that

leN(Q)- EeN(Bj)1 < j

T.

61

3.1. THE CARATHEODORY CONSTRUCTION

The set T = Q \ (UjBj) is still compact and has Lebesgue measure less than T. By the outer regularity of Lebesgue measure, we may find an open set W ;2 T that has measure not exceeding 2T. Next, we may cover T with finitely many balls CIt, of radii less than 6, and lying in W. By the covering theorem of Lebesgue (see Section 3.5 or Hurewicz and Wallman [1]), we may assume (by passing to a subcovering) that no point of T is covered more than N + 1 times. In summary, the balls Bj,C" are finite in number, have radii less than 6, and cover Q. Thus we have an upper bound for Sf(Q) given by ETN[idiam(Bj)]N + ETN[!diam(C,,)]N j

"

=

E.eN(Bj) + E.eN(C,,)

~

.eN (Q)

j

"

+ (N + I) . .eN (T)

< .eN(Q)+(N+I)·T. Since T was arbitrary, we find that Sf(Q):5 .eN(Q) = 1. On the other hand, if the balls B j have radii less than 6 and cover Q then j

j

Thus Sf(Q) ~ 1. Combining our two estimates yields that Sf (Q) = 1. Letting 6 -+ 0+ yields that SN (Q) = 1. _ Hausdorff Dimension Another measure closely related to spherical measure is the Hausdorff measure. The a-dimensional Hausdorff measure, denoted by 1£01, is the result of Caratheodory's construction from the family of all bounded nonempty subsets of aN and the gauge '/r 0l/2

O'OI(A) = 2 ar(a/2)

(diam(A») 2

Q

(3.10)

Since this gauge is an extension of 0'01 to a larger family of sets, it is appropriate to use the same notation. It also follows immediately, from the fact that Hausdorff measure is defined by extending the gauge to a larger family of sets, that 1£01 :5 SOl • Any non-empty bounded set A is contained in an open ball with radius larger than, but arbitrarily close to, the diameter of A. It follows from this observation that SOl :5 201 1£01. In particular, we know that spherical measure and Hausdorff measure have the same sets of measure zero.

CHAPTER 3. MEASURES

62

Calculations of the sort made in Example 3.1.8 can be used to show that if E is a polyhedron, or more generally the closure of a domain with C 1 boundary, then the SN measure of E, the llN measure of E, and the Lebesgue measure of E are all equal. See Federer [4]. Definition 3.1.9 Let E eRN be V(E)

= =

II

compact set. Let

inf{a > 0 : llQ(E) inf{a > 0 : SO (E)

= O} = OJ.

We call V(E) the Hausdorff dimension of the set E. Remark 3.1.10 IT E C aN is compact and V(E) > 0, then we also have V(E)

= =

sup{a > 0: llQ(E) = co} sup{a > 0: SQ(E) = co}.

In an informal treatment of the Cantor ternary set, we usually add up the lengths of the segments in the complement and find that the sum is 1. So, naively, the Cantor set has length zero. The measures we have introduced allow us to make that calculation more rigorous. Now we may ask the more precise question: What is the Hausdorff dimension of the Cantor s e t ? " "" An elementary calculation, which we leave to the reader, determines that the Hausdorff dimension" of the standard Cantor set is In2/1n3. By varying the lengths of the intervals removed, and the rate at which they are removed, one can create Cantor sets of any Hausdorff dimension strictly between 0 and 1 (see Falconer [1], Section 1.5). Although variants of the ideas presented here have reappeared in the mathematical literature in modern guise, and have been dubbed ''fractal dimension", it should be noted that the original idea is due to F. Hausdorff and is at least eighty years old. Exercise: IT MeaN is a CI, compact hypersurface, in particular if it is the boundary of a bounded domain, then the Hausdorff dimension of M is precisely N -1. [Hint: First, the problem is local so work on a coordinate patch in the boundary. Also, the problem is invariant under diffeomorphisms so map the coordinate patch to a piece of JRN-I. But in the examples above we have already (implicitly) determined the Hausdorff dimension of aN-I.] Other Measures The general scheme of Caratheodory's construction can be usefully applied for many choices of the family of subsets :F and of the gauge u.

3.1. THE CARATHEODORY CONSTRUCTION

63

Definition 3.1.11 The following measures are defined on lRN by applying CaratModory's construction: (i) The a-dimensional Hausdorff measure, 1£a, a f'eIUlts when

eRn {t: t

:F is the family of all subsets of aN and U =

Ua

7r a / 2

= 2 ar(a/2)

(diam(A») 2

(ii) The a-dimensional spherical measure, results when

a

(3.11)

(3.12)

sa, a eRn {t : t ~ OJ,

:F is the family of all open balls in lRN Ua

OJ,

where

ua(A)

and U =

~

(3.13)

is as in (i).

(iii) The m-dimensional integral geome.tric measure with exponent t, z;n, mE {O, 1,2, ... }, t e {r: 1 ~ r ~ oo}, results when

:F is the family of all Borel subsets of lRN and U =

Um,t

(3.14)

where

(3.15)

with p. the invariant measure on the space of orthogonal projections from aN to Rm [Federer 4, §2.7.16(6)] and with Cm,t an appropriate normalizing constant, and

um,oo(A) is the essential supremum of .cm[P(A»),

(3.16)

with .cm[P(A») considered as a function on the space of orthogonal projections from aN to am with the invariant measure. (iv) The m-dimensional Gross measure, gm, m e {O, 1,2, ...}, results when :F is the family of all Borel subsets of aN (3.17)

= uG,m where uG(A) = sup {.cm[P(A»):

and U

p orthogonally projects IItN to

am}. (3.18)

64

CHAPTER 3. MEASURES

(v) The m-diInensional convex integral geometric measure with exponent t, Qr', m e {0,1,2, ...}, t e {r: 1 ~ r ~ oo}, results . when

:F is the family of all open conveI: su.baetl of R!i and U

(3.19)

= Um,t is aa in (iii).

(vi) The m-dimenaional convu integral geometric meuu.re with exponent 1, Qr, is also known aa the m-dimensional Gillespie measure. (vii) The m-dimenaional conveI: integral geometric meuu.re with exponent 00, Q:;', is also known aa the m-dimensional Caratheodory measure. (viii) The m-dimensional symmetrized Hausdorffmeasure, {O,1,2, ... }, reaults when

rm, m e

:F is the family of all subsets 01 r' and U

= US,m

(3.20)

where

us,m(A) = 2- m T m sup{ m-dimenaional area olm-dimensional pamllelepipeds formed from vectors Pi - qi, i = 1,2, ... m, with Pi,qi e A} (3.21) . Remark 3.1.12 While all the measures defined in Definition 3.1.11 give the same value on any open piece of a smooth surface and all coincide with Lebesgue measure if N = m, they are not all equal on arbitrary sets. The interested reader should see the references cited in Federer [4J, §2.1O.6. Definition 3.1.13 For measures /A and v on JR.N, we will write (i) /A9!v

il and only if there exist non-negative real constants that holds lor all A C

]RN ,

(ii)

il and only if /A(A) = 0 {:=} v(A) = 0

lor all A C

]RN .

Cl

and

C2

such

3.1. THE CARATHEODORY CONSTRUCTION

65

We collect the relevant facts in the next theorem. Proofs and examples to substantiate these claims can be found in Federer [4], §2.10.6 and the references cited therein. . Theorem 3.1.14

(i) For m

e {O, 1, 2, ...}, 1 :::; t < co, S'"

~

'H.m

~

rn ?

Q: ~ gm ~ z: ? z:" .

(ii) For me {O, 1, 2, ...}, (iii) Form

e {O,I,2, ...}, s,t E {r:

1:::;

r:::; co},

(iv) Form e 1,2, ... , the measures defined in Definition 3.1.11 form the following three equivalence classes under the relation ~

{sm, lI.m, rn} U {Qr' : 1 :::; t :::; co} {am}

{z:" : 1 :::; t

:::; co}.

(3.22) (3.23) (3.24)

In general, the interaction of mappings and measures can be complicated, but for the two most important measures in Definition 3.1.11 certain facts follow easily by considering the e~ts of the mapping on ,the gauge.

Definition 3.1.15 A function f : X -+ Y, with X and Y subsets of Euclidean spaces, is a Lipschitz function 1 if there exists C < co such that

The Lipschitz constant of the Lipschitz function f is the smallest number C such that (3.25) holds. The Lipschitz constant of f is denoted by Lip(f). Theorem 3.1.16 Suppose

Q

E IR n {t : t ~ OJ. If I: IRN -+ IRM satisfies

I/(x) - l(y)1 :::; L Ix - yl,

il I is a Lipschitz function with Lipschitz constant not exceeding L, then lor any A C lllN

i.e.

1 Note that our Lipsc:hitz functions are a special case of the Lipschitz or Holder functions of harmonic analysis (see Krantz [1».

CHAPTER 3.

66

MEAS~

Proof: For Hausdorff measure, note that the diameter of the / image of any set A is bounded by L times the diameter of A. For spherical measure, note that the f image of the sphere centered at P with radius r is contained in the sphere centered at f(P) with radius L r. • The following special case of the theorem is so important that we state it separately. Corollary 3.1.17 Suppose a e III n {t : t ~ OJ. 1/ IT : RN -+ aM is an orthogonal projection or orthogonal injection, then for any A C JlN 1la(II(A)] :::; 'M°(A),

3.2

sa[IT(A)]:::; S'"(A).

(3.27)

Rectifiability

Definition 3.2.1 By an arc is meant a homeomorphic image of some closed internal [a, b] C III 1/ r is an arc in ]RN, say given as the homeomorphic image of [a, b] under "'( : [a, b] -+ RN, then the length of r, denoted len(r), is given by len(r)

= sup {f)y(ti- 1) -

~(ti)1 : a = to < tl < t2 < ... < tm = b} .

•=1

, (3.28) If the length of an arc is finite, then the arc is said to be rectifiable. Remark 3.2.2 Of course, the length of an arc may be infinite. Early in the twentieth century, W. F. Osgood was motivated by considering the boundary behavior of conformal, mappings to construct an example of an arc with positive area in the plane (see Osgood [1]). In fact, examples show that an arc in lItN may have positive N-dimensional Lebesgue measure. (See Parks and Schori [1] for a construction based on the heuristic principle that given a Cantor set one can always run an arc through every point of it.) Note that because an arc is required to be the homeomorphic image of an interval, examples of space-filling curves (see Section 3.4) tell us nothing concrete about how badly behaved an ,arc may be. Lemma 3.2.3 If r is a rectifiable arc in RN , then r can be pammetrized by arc length, meaning that with l len(r), there exists a homeomorphism "'(0 : [0, l) -+ RN such that for each 0 < s :::; l the image of [0, s] has length

=

s. Proof: Let "'f : [a,b] -+ RN be the given parametrization of r. Define 'Yo(s) "'(t) where t e [a,b] is that unique number such that

=

len ( ",(([a, t]))

= s.

•

3.2. RECTIFIABILITY

67

Theorem 3.2.4 If r is an arc, then len{r)

=?l1(f).

(3.29)

Proof: By the fact that orthogonal projection onto a line in liN does not increase l-dimensional Hausdorff measure, we can see that ?l1(r) ~ len(r). The reverse inequality holds because the arc-length parametrization of a rectifiable curve is a map from R to aN with Lipschitz constant 1. • The preceding results show that the natural definition of length arising by considering inscribed polygons is well-behaved and agrees with the 1dimensional Hausdorff measure. The situation in highe~ dimensions is far more complicated as the fo11owing example due to H. A. Schwarz shows (see GelbaUJl!..and Olmstead [1) for more detail). Example 3.2.5 The right circular cylinder of radius 1 and height 1, thus with area 211", can be approximated by a family of 4mn, m, n E N, congruent inscribed triangles determined by (2m + l)n points . Pi" =

{

sin~ ..iL) ( cos~ n' fa '2m

(

COS

keven,

i)

(2j+1)".. (2Hl),.. n ,sm n ' 2m

k odd.

lying equally spaced on 2m + 1 equally spaced cross-sectional circles. The area of this polyhedral approximation is sin~1 ( 11")' 211" ; n V1 + 4m2 1 - cos;; '.

(3.30)

While the diameter of the triangles goes to zero whenever m and n approach infinity, the limit of the expression in f3.30) can approach any value in the interval [211",00] depending on how m and n grow. The example of Schwarz shows that the area of a surface cannot be computed by taking the supremum of the areas of all possible polygonal approximations to the surface. The example does indicate that the limit inferior of the areas of approximating polyhedra may define the appropriate value for the area. Following Federer [2], we give the definition of Lebesgue area that is the appropriate extension of the notion of the length of an arc. Definition 3.2.6 (i) A set of m + 1 points AI,Pl, ... ,Pm in liN is said to be in general position if. for any set ofm+ 1 real numbers Ao, ~1"" ,~m, we have m

o= E~i i=O

m

and 0= L~iPi imply 0 = ~o = ~1 = ... = ~m. i=O

Equivalently. the points are in general position if and only if PI Po,p, - Po,··· ,Pm - AI are linearly independent.

68

CHAPTER 3. MEASURES

(ii) Given m + 1 points Po,Pl, ... ,Pm in RN in general position, the msimplex determined by those points is denoted fPo,Pl, ... ,Pm1 and is defined ~y setting

The points PlhPl, ... ,Pm are said to be the vertices of the simplex. The m + 1 numbers.\o, ~1" •• ' ~m are called the barycentric coordinates 0/ the point E~ ~iPi.

(iii) For a given m-simplex fPo,Pl, ... ,Pm1, and for each integer 0 ~ k < m, we call each k-simplez, fpio,Pil" .. ,pi.1,.a face of the simplex fPo,Pb··· ,Pm1·

.

(iv) A simplicial complex in]iN is a set C 0/ simplices such that (a) every face of any simplex in C is also in C, (b) the intersection of two simplices in C is either empty or a face 01 both 01 them, and (c) each compact subset of]RN intersects only finitely many simplices in C. . Definition 3.2.7 (i) A finite triangulation of a topological space X in a Euclidean space ]Rm is a pair (K, T) where K us a finite simplicial complex in]Rm and T is a homeomorphism mapping X onto the union 01 all elements of K. In this case, X is said to be finitely triangulable

(ii) A function I from the topological space X to RN is quasi-linear il there e$ists a finite triangulation (K, T) in some Euclidean space Rm and there exists a linear function L : Rm -+]RN such that I = LOT. (iii) II X is finitely triangulable and I : X -+ RN is continuous, then the k-dimensional Lebesgue area of f, denoted LA,,(f), is the infimum 01 all the real numbers t such that: For each

€

> 0 there exists a quasi-linear function g : X -+ JRN with Ig(x) - l(x)1 ~

€

lor all x EX

and

where N(g,y) is the cardinality N(g,y)

01 g-l(y),

i.e.

= card {x EX: g(x) = V}.

69

3.2. RECTIFIABlLITY

Strictly speaking, the k-dimensional Lebesgue area is a number associated with a function rather than a set, but the terminology is often abused by referring to the k-dimensional Lebesgue area of the set that is the image of the function. For smooth surfaces, the k-dimensional Lebesgue area agrees with the standard value of the k-dimensional area computed using multi-variable calculus, but unfortunately, even this more sophisticated definition of area comes to grief as is shown by the next example. The point of this example is that a set in Jll3 with finite Lebesgue area may also have positive 3-dimensional Lebesgue measure.

Example 3.2.8 (Besicovitch [2]) Let A be the closed unit cube in Jll3. We'can subdivide A into 8 congruent subcubes and inside each of these we can construct another closed concentric cube with sides parallel to the sides of A, but having volume i . ~. These slightly smaller disjoint closed cubes will be denoted by At,A2 , ••• ,As. Now considering Ai, i = 1,2, ... ,8, we can similarly subdivide Ai into 8 congruent subcubes and inside each of these we can construct another closed concentric cube with sides parallel to the sides of A,' but having volume l· b· These cubes will be denoted by Ai.t, A i •2 , •• ·, A i•S • Supposing Ail.i2 .....i. with volume ~t~: ./ro to have been constructed, we subdivide it into 8 congruent subcubes and inside each of these construct another closed concentric cube with sides parallel to the sides of A, but . volume 1±1Zt I .. haVIng 2 • ij7i+T' We denote those cubes by A i1 .i2 •...• i .,j, for j =.1,2, ... ,8. Setting A·· It,12,···tl.• ,

u

we see that the volume of Dk is ~t1: and DI ::> D2 :J ... :J Dk ::> ••• , so we may set D=nDIo k

and conclude that D is a compact set with Lebesgue measure ~. We will construct a continuous function f from the unit square in ]R2 into the unit cube A such that D is in the image, but the Lebesgue area of f is finite. Choose a closed square B on some side of A and in the relative interior of B choose 8 pairwise disjoint closed squares Ct, C2,"" Cs. On each Ai, i = 1,2, ... ,8, choose a closed square on some side of Ai and denote it by Bi . Clearly, it is possible to construct non-intersecting polyhedral surfaces PI, P2 , ••• , Ps such that each Pi is homeomorphic to a right circular cylinder of finite height and has as its boundary arc the edges of Cj and the edges of B i • Further we can require that, except for the edges of C i , Pi is interior to .A and, except for the edges of B i , Pi is exterior to D I . Note that the area

70

CHAPTER 3. MEASURES

of each Pi can be made arbitrarily small by choosing small squares, so that the cylinder is thin. Indeed, the cylinders at each stage can be chosen to have small enough area that the total area of the union of all the cylinders is bounded above by any positive number given in advance. To construct the continuous map, let B' be the unit square in a2 • Let q, c~, ... , C~ be disjoint squares interior to B', and for i = 1,2, ... , 8, let Bi be a concentric square to Ci that is interior to C;. One can construct a quasi-linear map (see Definition 3.2.7(ii)) from B' \ UiC; onto B \ UiCi, from C: \ Bi onto Pi, and from Bi onto Bi • The construction is extended by repeating the above with A, B, and B' replaced by Ai> B i , and Bi. The mapping extends to the entire unit square by continuity, and its image must contain all of D. But also the construction of the mapping may . be terminated as a quasi-linear mapping at any stage by mapping each Bi1,i, •.... i .. onto B iloi••...• i ... Thus the Lebesgue area of the image is the sum of the areas of the cylinders, hence is finite. Note that despite the fact that f has finite Lebesgue area, the image of f is a set with 3-dimensional Lebesgue measure ~. The upshot of the investigation of various phenomena in higher dimensions is to put the measure theoretic approach of Caratheodory in a commanding position. T:he various measures defined in Section 3.1 differ from each other on certain sets and agree with ·each other on more well-behaved sets. To describe the various classes of sets requires some definitions, The proofs of the main results are extremely. technical, so we will state such results without proof and refer the interested reader to Federer [4]. Definition 3.2.9 Suppose sCaN, cp is. a measure on RN, and m is a positive integer. . (i) S is said to be m-rectifiable if S is the image of some bounded subset of Rm under a Lipschitz function.

(ii) S is said to be count ably m-rectifiable if it is a countable union of m-rectifiable sets or equivalently the image of some subset of Rm under a locally Lipschitz function. (iii) S is said to be countably (cp, m)-rectifiable if there is some countably m-rectifiable set that contains cp-almost all of S. (iv) S is said to be (cp,m)-rectifiable if it is countably (cp,m)-rectifiable and tjJ(S) < 00. (v) S is said to be purely (tjJ, m)-unrectifiable if S has no m-rectifiable subset A with cp(A) > O. All the m-dimensional measures of Section 3.1 agree on the m-rectifiable sets; indeed, a measure on Euclidean space that does not assign the same

3.2. RECTIFIABILITY

71

value to an m-rectifiable set as does 1lm cannot reasonably be caUed. an m-dimensional measure. The next result shows us that the general agreement on the appro:.· . priate value for the measure of an m-dimensional set extends beyond the m-redifiable sets. Theorem 3.2.10 Let m be a positive integer and suppose 1 ~ t is an (1l m , m) -rectifiable subset 01 aN, then S"'(A)

= llm(A) = m(A) = g"'(A) = Q;"(A) =zr'(A).

~ 00.

II A

(3.31)

As a consequence of Theorem 3.2.10, the class of (1lm,m)-rectifiable subsets of aN has a special place in the study of geometry and analysis in Euclidean space. The next theorem shows how close to Cl such sets are. Theorem 3.2.11 Let m be a positive integer. A subset o/aN is countably (ll m , m) -rectifiable il and only if, except lor a set 0/ ll m measure zero, it is contained in a countable union 0/ C 1 submanilolds 0/ aN.

One of the deepest theorems in the subject is the Structure Theorem for Sets of Finite Hausdorff Measure. Originally proved in Besicovitch [1] for I-dimensional sets in ]R2, it was extended to general m and N in Federer [1]. Theorem 3.2.12 (Structure Theorem) Let m be a positive integer and suppose 1 ~ t ~ 00 • .11 A is a subset o/]RN with finite m-dimensional Hausdorff measure, then there exists a countably m-rectifiable set R C ]RN such that

(i) A \R is purely (1l m ,m)-unrectifiable,

(il) If{A \ R) =

o.

No easy proof of the Structure Theorem is known. The existing proofs involve careful study of the behavior of AnX(a,r, V,s)

where A is the given set of finite Hausdorff measure, a E aN, 0 < r < V is an m-dimensionallinear subspace of]RN, 0 < s < 00, and X{a, r, V, s)

= lIi(a, r) n {x : dist(x -

a, V) < six -

00,

all.

Another important tool used in the study of lower dimensional sets is the notion of the density of a set: Definition 3.2.13 Suppose A C

JRN.

(i) The a-dimensional lower density of A at x is denoted by ~~ (A, x) and is defined by setting ......Q(A

..r.

) _lim·nf 1la

, X-I

r~O

(AnB(x,r») l' . ara

(3.32)

12

CHAPTER 3. MEASURES

(il) The a-dimensional upper density of A at:J: is denoted by ()·Q(A,:J:) and is defined by setting .......Q(A 'M'

(iii) In case

()~(A,:J:)

,:J:

)

1{.Q (Ani(:J:,T») = Iimsup l' . r.j.O QTo

(3.33)

= ()·Q(A,:J:) we set

()Q(A,:J:)

= ():(A,:J:) = C).O(A,:J:).

(3.34)

We call ()Q(A,x) the a-dimensional density of A at :J:. Theorem 3.2.14 Suppose 0 5 Q. For any 1{.0 measurable set A C liN fllith 0 < 1{.O(A) < 00, 2- Q 5 ()·Q(A,:J:) (3.35) holds fOT 1{.Q almost every :J: E A. Proof: Arguing by contradiction, suppose A is 1{.0 measurable and

< 2-"'}

{:J:: C)·Q(A,x)

has positive 1{.Q measure. We conclude that there exists 0 < t < 2- Q, 0< TO < 00, and a 1{.Q. measurable set B C A .with 0 < 1{.Q(B) < 00 such that (3.36) holds for :J: E B and 0 < T < TO. Let 1 < ,x be arbitrary. By the definition of Hausdorff measure, there is a covering of B by countably many sets S; of diameter less than TO such that 00 QT Q(diam(S;»0. >"1{.Ct(B) > (3.3.1)

L2;=1

We may assume that each set Si contains some point :J:; of B. Let us also set T; = diam(8i ). Then, by (3.36) and (3.31), we have 00

,x1{.Q(B)

>

L2-

Q

T Q(Ti)Q

;=1 00

~

L 2- Qt- 1{.Q (B n j(Xi, Ti» 1

i=1 ~

2- Qt- 11{.Q(B).

Since 0 < 1{.Q(B) < 00, we conclude that>.. > 2- Ct r since the choice of 1 < ,x was arbitrary, we have t

~

2-

Q

1

or

t,x >

2- Q. But

,

contradicting the choice of t. The following amazing theorem holds for the densities:

•

3.2. RECTIFIABILITY

73

Theorem 3.2.15 (Marstrand) Suppose A C lllN is

1{.0I.

measurable and

o < 1lOl.(A) < 00. If .. o < ~OI.(A,x) < 00 for all x e A,

(3.38)

then a is an integer. The complete proof of this theorem is very hard. Following Falconer [1], we present an easy proof of part of the result. Proof that a f/. (0,1) : Arguing by contradiction, suppose 0 < a < 1. For each 1 < 'I, we consider an annular density

:ia(A ) -1' 1£01. (A n iii(x, qr) \ 1II(x, .,.. ,X,'1 - 1m T OI.rOl. r~O

rn .

Because the density exists at every point of A, we see that a simple difference of limits gives us

for each x e A. Now, by Theorem 3.2.14, we can find ro > 0 such that X

1

= {x E A:"2 ~

1l0l. (A n iii(x

TOI.rOl.'

rn

V0

< r < ro

}

has positive 1la measure. Let Xo be an accumulation point of X. For points of x EX \ {xo} set

rex) = 21x --~01/(1 + 1/). Then iii(x, ('1-1)r(x)/2) C iii(x, 1/r(x» \ 18(x, r(x», so, if rex) is sufficiently small, To. (1/ ;l~aar(x)a

~ 1{.0I. (A n iii(xO,'1r(x» \ 18(xo, r(x»)

holds. As x -+ Xo we find

(1/2~+~a ~ iOl.(A,xO,1/) = ('101. -l)~OI.(A,xo). Thus (3.39)

holds for every 1 < 'I. The left-hand side of (3.39) has an infinite limit as 1/ .J. 1, but the right-hand side is finite by hypothesis. This contradiction • proves the special case of the theorem.

lJl1A.PTER 3. MEASURES

3.3

Minkowski Content

Definition 3.3.1 Suppose A C]RN and 0 ~ K ~ N. The K-dimensional upper Minkowski content of A, denoted by M~K (A), is defined by

.eN {x : dist(x, A) M .K(A). =hmsup T N-K

< r}

N-Kr

rJ,O

Similarly, the K -dimensional lower Minkowski content of A i6 denoted by M!< (A) and defined by K() M. A

..

= hmrJ,Omf

.eN {x: dist(x,A) < r}

T N-Kr N

K

In case the K -dimensional upper Minkowski content and the K -dimensional lower Minkowski content of A are equal, then their common value is called the K -dimensional Minkowski content of A and is denoted by MK (A). We leave it to the reader to verify the following easy result. Proposition 3.3.2 Fix 0

~

K

~

N.

(i) If the sets A and B are sepamted by a positive distance, then M·K(AUB)

=

M!«AUB)

=

M· K (A) + M· K (B), M!, (A) + M!, (B).

(ii) The K -dimensional upper Minkowski content is a finitely sub-additive set function. Unfortunately, the finite sub-additivity described in the preceding Proposition does not extend to countable additivity (consider K > 0 and a countable dense set of points). Thus, the Minkowski content is not a measure, and in general the relationship between Minkowski content and the commonly used measures of K -dimensional size such as the Hausdorff measure is not very good. Clearly, the Minkowski content of a set and its closure is the same, but that is not the fundamental difficulty, for there are known to exist compact sets in JR3 with finite 2-dimensional Hausdorff measure and infinite Minkowski content (see Federer [4], §3.2.40). On the other hand, the next proposition gives us a general result bounding the Hausdorff measure by a multiple of the lower Minkowski content. Proposition 3.3.3 For 0 ~ K ~ N and any set A C aN, we have

l£K (A)

~ 3K T N-:':;' T K M~ (A).

(3.40)

3.3. MINKOWSKI CONTENT

75

Proof: We may assume that M~ (A) < 00. Let E > 0 be arbitrary. Then there are arbitrarily small positive numbers r such that

.cN{x:dist(x,A)
(3.41)

Fixing such an r, let J8(ai,r), i = 1,2, .. . ,n, be a maximal pairwise disjoint family F of open balls of radius r centered in A-maximal in the sense that there is no open ball of radius r centered in A that is disjoint from the balls in F. By (3.41), we have n $

.cN{x: dist(x,A)
(3.42)

We also have n

A

c

UB(ai,3r),

(3.43)

i=1

else there would exist another open ball of radius r centered in A and disjoint from all the balls in the maximal family. It follows from (3.42) and (3.43) that the 6r approximation to the K-dimensional Hausdorff measure is bounded by n TK (3r)K ::; 3K TNT:T K [€ + M~(A)].

First letting r .!. 0 and then using the fact that that the result follows.

€

(3.44)

> 0 was arbitrary, we see .

•

As mentioned above, there is no bound for the K -dimensional Minkowski content of a set by a multiple of its K -dimensional Hausdorff measure. The best result available is the following. Theorem 3.3.4 (Federer [4] §3.2.39) Suppose 1 ::; K ::; N is an integer. If A is closed and A C f(1RK) for some Lipschitz function f : ]RK -t ]RN, then MK (A) = llK (A). The proof of this result is fairly technical. It depends on a key lemma, Federer [4], §3.2.38, which bounds the upper K-dimensional Minkowski content of the Lipschitz image of a compact subset of]RK by the integral of the K-dimensional Jacobian (the K-dimensional Jacobian will be defined in Section 3.8). The difference in complexity between dealing with the image of a Lipschitz map and the image of a C1 map is very significant, and it is perhaps more instructive to consider instead the proof of the following simplified form of that key lemma.

CHAPTER 3. MEASURES

76

Proposition 3.3.5 Suppose I : RK -+ JR.N is Cl and A is compact with A

c {x: IDI(x}1 S

p},

. then

M-K[J(A)] S ~ £K (A). Proof: For 0

< t, set

aCt}

= sup{IDI(x) -

DI(a)1 : a

e

A,

Ix - al S

t}.

Since I is Cl, we see that (1 is monotone increasing and continuous with aCt) .,J.. 0 as t .,J.. O. Also we have I/(x) - l(a)1 $

Ix - al a(lx - aD,

(3.45)

whenever a e A. Fix a number 0 < .\ < 1. For any given sufficiently small positive r we can find a positive h such that . (3.46) Clearly, we have h .,J.. 0 as r .,J.. O. With r given and h chosen as in (3.46), divide IRK into cubes of side length h. Thus we may cover A by a family of closed cubes Ilth, ... ,In , all of side length.h which intersect only on their faces, such that n

U I; C {x: dist(x,A) S vKh}.

;=1

We may and shan suppose each cube contains a point of A. We eaSily estimate the number, n, of cubes by (3.47) Now suppose a E A n I;, and consider the image of I; under the affine map, g, sending x to I(a) + (x - a,DI(a». We see that g(I;) is a K -dimensional parallelepiped with side lengths not exceeding ph. Further, we have I(I;) c {x: dist(x,g(l;» S vKha(vKh)}, and {x: dist(x,/(I;»

< r} C {x: dist(x,g(I;}) < r + vK h(1(vKh)}.

For a K-dimensional parallelepiped, P, in RN with side lengths bounded by s it is elementary to estimate K-l

.cN{x: dist(x,P)
(3.48)

3.3. MINKOWSKI CONTENT

77

where the Ci are constants. Applying (3.48) to g(I;) with

8

ana t replaced by r + VKhu(VKh), we see that .eN {x: dist(x,g(I;» < r

replaced by ph

+ v'Khu(v'K h)} =

K-l

L Cdph)i (r + VKhu(v'Kh»N-i i=O

Thus we have

£N {x: dist(x,/(I;» < r} ~ K-l n ( Ci (ph)i (r + VKhu(v'Kh»N-i

L

i=O

Dividing (3.49) by T N-K r N- K and using (3.46) and (3.47), we estimate

£N{x:dist(x,/(I;»
.

( ~ Ci pi hi rK -i (r + v'K hu(.v'K h»N-i ~ TN_KhKr N-,

+

TN-K pK hK (r +v'Khu(v'Kh»N-K) TN_KhKrN- K .

(3.51)

By (3.46), we have

(r+VKhu(VKh»N-i r N -,.

for i

-t

1

10 asr+ ,

(3.52)

= 0,1, ... , K. But (3.46) also implies (3.53)

=

for i 0, 1, ... , K -1. The result now follows from (3.51), (3.52), and (3.53) since

It is interesting to observe that the Minkowski contents are well-behaved .relative to cartesian products.

78

CHAPTER 3. MEASURES

Theorem 3~3.6 1/ A

c aM

and B C

ItN, then

2-[(M+N)-(K+L»)/2 TM":'KTN-L M!,(A) . M~(B) ~ T(M+N)-(K+L) M~K+L)(A x B) ~ M*(K+L)(A x B) ~ TM-kTN-L M*K(A) • M*L(B). T(M+N)-(K+L) Proof: This follows from Fubini's Theorem since a point (x, y) within distance'r of Ax B must have x within distance r of X and have y within distance r of y. Similarly, if x is within distance r of some point of A and y is within distance r of some point of B, then (x,y) is within distance v'2r of a point of Ax B. • The upper Minkowski dimension of a set A can be defined to be the infimum of the numbers K such that M*K(A) = 0, or equivalently the supremum. of the numbers K such that M*K (A) > O. A similar definition is made for the lower Minkowski dimension. Unfortunately, these dimension!! do not have the fundamental property that the dimension of a countable union of sets is the supremum of the dimensions of those sets. This is another reflection of the fact that the Minkowski contents are not measures. We can make an additional definition to create a countably sub-additive set function based on the upper Minkowski content, and thus we will obtain a measure in the usual sense. Definition 3.3.7 Suppose A C RN and 0 :5 K :5 N. The K -dimensional upper Minkowski measure of A, denoted MK (A), is defined by setting MK (A) = inf {

L M*K (B) : F C <jlN, card(F) ~ No,

A C UF} .

BeF

(3.54)

Similarly, the K -dimensional lower Minkowski measure of A is denoted by MK (A) and defined by MK (A) = inf {

L

M!, (B) : F C <jlN, card(F) :5 No, A C UF} .

BeF

(3.55)

We leave it to the reader to verify the following easy result. Lemma 3.3.8 For each 0 additive.

~

K

K :5 N, -K M and M are countably sub-K

K

Remark 3.3.9 The preceding lemma shows us that M and M are outer measures. The issue of which sets are measurable is settled by the next proposition.

3.3. MINKOWSKI CONTENT

79

Proposition 3.3.10 All open sets of]RN are MK and MK measurable, and both MK and MK are Borel regular. Proof: This is a consequence of Caratheodory's criterion, Federer [4], §2.3.2(9), and Proposition 3.3.2. To see this, suppose that A and B are separated by a positive distance and let f > 0 be arbitrary. Let F be a countable cover of A U B such that f

+ M(A U B) ;?!

E M*K (C). CeF

Then ;?!

E M*K (C n (A U B» CeF

=

EM*K«CnA)U(CnB» CeF

=

E M*K(CnA) + E M*K(CnB)

CeF

;?!

Since £

MK (A)

CeF

+ M(B).

> 0 was arbitrary, we have

as required. The argument for MK is similar. -__ To see that the measure MK (respectively, MK) is Borelregular, simply use the fact that "the upper (respectively, lower) Minkowski content of a set and its closure are equal. • To relate these measures to each other and to other quantities we have the following result. Proposition 3.3.11 Fix 0::; K ::; N. Then the following hold. (i) MK ::; MK ::; M*K .

(iii) If K is an integer, then the K dimensional lower Minkowski measure is bounded below by the K dimensional Gross measure (see Definition 3.1.11), that is, gK ::; MK.

CHAPTER 3. MEASURES

80

(iv) If A is contained in the image of a compact subset of]RK under a Lipschitz junction, then

Proof: (i) and (ii) Easy exercises. (iii) Recall that the K dimensional Gross measure results from Caratheodory's construction using as the gauge the supremum of the Lebesgue measure of the orthogonal projection onto]RK. Since Fubini's Theorem tells us that holds for any orthogonal projection II : ]RM -+

jRK,

it is easy to see that

M!,< (S) :::: gK (5) must hold for any 5 C JRN. Then it follows that for any countable cover F of a set A we have

gK (A) :$

L

. BEF

gK (B) :$

L

M!,< (B),

BEF

.

and the inequality gK (A):$ MK (A) follows by takiilg the infimum over all such countable covers of A. (iv) By Federer [4], §3.2:26 and §3.2.39, we know that gK (A) = 1I.K (A) = MK (A). Thus we have

1I.K (A)

= gK (A)

:$

MK (A) :$ MK (A) :$ M*K (A)

= 1iK (A).

•

Corollary 3.3.12 In JRN, the measures M N , M N , and eN agree. Proof: All three are translation-invariant measures that assign the same measure to the unit ball. • Remark 3.3.13 Note that it is a consequence of the corollary that a set in llN with Lebesgue measure zero can be covered by countably many sets whose upper Minkowski contents add to less than any given positive number. Using the measures MK and M K , we define two more notions of dimension. To agree with the literature, we call them packing dimensions.

3.4. A SPACE-FILLING CURVE

81

Definition 3.3.14 Suppose A C lIlN. The upper packing dimension A, denoted by dimp A, is defined by

0/

dim" A

= inf { K : MK (A) = O} = sup { K

Similarly, the lower packing diJnension defined by

dim" A

0/ A

: MK (A)

> O} •

(3.56)

is denoted dim" A and is

= inf {K :MK(A) = O} = sup {K: MK(A) > O}.

(3.57)

The interested reader should see Mattila [1] for much more information on these matters.

3.4

A Space-Filling Curve

The first example of a space-filling curve was presented in Peano [1]. The existence of such a curve was startling, and to this day remains surprising to those first learning of it. Typical constructions are quite geometric as in Gelbaum and Olmstead [1], but Peano's original construction was·not. Here we give another interesting non-geometric construction. For motivation we define a map, 9 : [0,1) x [0,1) -+ [0,1) as follows: Given (x, y) E [0,1) x [0,1) express x and y as non-terminating decimals which do not have an infinite sequence of 9's. Thus we have . = Y =

x

0.CIC2 ••. ,

0.d1 d 2

••. ,

and for any integer n there exist n' and nil such that n' > n, nil > n, c,.. t= 9 and dn.. f;. 9. Then g(x, y) will be the real number with decimal expansion Lemma 3.4.1 The map g: [0,1) x [0,1) -+ [0,1) is one-to-one.

Proof: This is obvious, since the decimal expansion of g(x, y) cannot have an infinite sequence of 9's. • One would like to show that the inverse of 9 is continuous where it is defined, but unfortunately that is just not true: To see this, let the inverse of 9 be denoted by / and consider the two numbers s and t s

t

= =

0.7 {2n 9's} 00 .. . 0.8 {2n D's} 00 ... ,

82

CHAPTER 3. MEASURES

the difference between which is 10-2n -

1•

Both s and t are in the image of

g, and we have

J(s) J(t)

=

(0.7 {n 9's} 00 ... , O. {n 9's} 00 ... ),

=

(0.800 ... , 0.00 ... )

the distance between which always exceeds 0.9.

•

The lack of continuity in I above is caused by the arbitrarily long sequences of 9's that may exist in the decimal expansions of elements of the image of g. To avoid arbitrarily long sequences of 9's, we will introduce an alternative representation of real numbers. Simply put, the real number in [0,1) is to be expressed in decimal form without an infinite sequence of9's, and then any finite sequence of k copies of the digit 9 is to be replaced by "9 {the binary expression for k} 9". We introduce the notation

*d1 d2d s ·· . for numbers in [0,1) expressed in this way. Note that the sequence of digits d1 ,d2,da, •.. cannot be arbitrarily given; the 9's must fall into pairs separated by binary expressions for positive integers. To illustrate, we have 0.9799799977 ... = *919791097911977 ... Using this new notation, we can now defi~e. G : [0,1) x [0,1) -+ [0,1) by writing

x Y

= =

*CIC2 ••• ,

*d1 d2 ... ,

and setting Note that the preceding expression for G(x, y) must be an ordinary decimal expansion, since we cannot guarantee the proper pairing of 9's. As before we have LemlDa 3.4.2 The map G : [0,1) x [0,1) -+ [0,1) is one-to-one. We denote the image of G by X, and let F : X -+ [0,1) x [0,1) be the inverse of G. LelDlDa 3.4.3 The maximal length Jor any sequence oj consecutive 9's in the decimal expansion oj any number in X is £. Proof: This is clear since neither of the expressions

=

*CIC2 ••• ,

Y =

*d1 d2 ···,

x

3.5. COVERING LEMMAS can have a pair of consecutive 9's.

83

•

L~mma

3.4.4 If two number., in X differ by le.,s than 10-", thf!1J thq mU8t agree in the first n - 3 decimal place8.

Proof: This is clear because there are at worst a pair of 9's occurring in the nih and (n - l)·t decimal places. • Lemma

3.~.5

F i8 uniformly continuoU8 on X.

Proof: The fact that numbers in X differing by less than 10-" must have the same first n - 3 digits insures that the inverse images must agree in each coordinate up to the first k digits where k is the greatest integer less than (n - 3)/2. • Theorem 3.4.6 F extends to a continuoU8 function mapping onto [0,1] x [0,1]

from [0,1]

Proof: Since F is uniformly continuous, it extends continuously to the closure of X. The further continuous extension to [0, 1] follows from.Tietze's Extension Theorem (see for example Dugundji [1]). Since F is continuous on [0,1], the image of F must be compact. Also F is an extension ·of G-l, so F maps onto [0,1) x [0,1). Thus we have [0,1] x [0,1]

3.5

= [0,1) x [0,1) C F([O, 1]).

•

Covering Lemmas

Preliminary Remarks

The technique of covering lemmas has become an entire area of mathematical analysis (see, for example, de Guzman [1]). It is intimately connected with problems of differentiation of integrals, with certain maximal operators (such as the Hardy-Littlewood maximal operator), with the boundedness of multiplier operators in harmonic analysis, and concomitantly with questions of summation of Fourier series. Our intention in this section is to give an overview of the geometric aspects of covering lemmas of various kinds-not with an aim for completeness, but rather to give the reader a taste of some of the basic covering lemmas and how they can be used. Along the way, we shall provide a variety of references for ancillary reading.

CHAPTERa MEASURES

84

The four basic types of covering lemmas on which we shall concentrate are the Wiener covering lemma, the Besicovitch covering lemma, the Lebesgue covering lemma, and the Vitali covering lemma. The Wiener and Vitali covering lemmas are 'of interest because of their interaction with measure theory. The Besicovitch and Lebesgue lemmas have a strong intuitive geometric appeal. They are also useful because they are independent of measure theory, and depend instead on critical geometric artifacts of space. Suppose that a compact subset KeRN is covered by a family of open sets. Our first reaction to this hypothesis is that, because K is compact, we may pass to a finite subcovering. However, in the theory of covering lemmas, we wish to find a subcovering that is "thin" in some sense. For instance, we could have a finite covering {U",} of K c R2 by open sets with the property that some points in K are covered 5 times (that is, by five of the Ua ) and other points in K are covered 1010 times. (The maximum number of times that any point is covered is called the valence of the covering.) This last seems somehow wasteful. One would like to think that, if a compact set in ]R2 is cove"red by open sets, then it could be done fairly efficiently-so that the valence is fairly small. Figure 3.3 shows a covering 'of the unit square in the plane that has valence 10. (Note also that no element of the covering in Figure 3.3 can be discarded without leaving some points in the unit square uncovered.) Clearly, with simple modifications the valence could be increased to equal any large positive integer. Thus, interpreted ipso facto, the quest posed in the last paragraph is not possible. But if we are willing to pass to a refinement of the given open cover, or to restrict to covers by certain fairly regular types of open sets, then in fact the sort of "efficient covering lemma" that we seek is possible. The first such lemma that we shall discuss is the so-called Wiener covering lemma. The Wiener Covering LeInIna We begin with a formal statement of the lemma: LeInma 3.5.1 (Wiener) Let K be a compact subset of ]RN. Let B = {B",}aEA be a covering of K by open Euclidean balls. Then there is a subcollection {Ba;}};l of the balls such that

(i) the balls {Ba;}};l are pairwise disjoint,

(ii) if we write Ba;

= B(P

a; ,

raj)' then the dilated balls 3Ba; == B(Paj , 3raj J

cover K.

Proof: Since K is compact, we may as well suppose at the outset that the collection {Ba} is finite. We write B = {Ba}~l' Now select Ba! to have radius as large as possible (chosen from among the M given balls)-if there are several balls with this same largest radius, then just pick one of them arbitrarily. Note that there is no problem in selecting Ba! since we are working with only finitely many balls.

3.5. COVERING LEMMAS

85

Figure 3.3: Covering with Valence 10 Having selected B o" we now select Ba, subject to the two conditions that (1) it be disjoint from BOl and that (2) it have radius as large as possible. Again, if more than one ball satisfies these two conditions, then just pick one of them arbitrarily. . Continue in the preceding fashion. 1£ BOll' Ba., ... Bop have been selected, then we select Bop+! to satisfy the two conditions that (1) it be disjoint from B o" Bo.,. .. , Bjp and that (2) it have radius as large as possible. Clearly thIs process must stop, because our universe of balls is just the finite collection {Bo}~=l. Let 8' == {BaJ}!:l be the subcollection of balls that results from our selection process. We claim that the subcollection 8' satisfies the conclusions of the lemma. First, by design, the subcollection 8' is pairwise disjoint. We claim now that the dilated balls {3Boj } cover K. Of course it will suffice for us to prove that these dilated balls cover each of the original balls {Bo} ~=1. Let Bo, be one of these latter balls. If it is equal to one of the selected balls B oj , then of course it is covered by the dilated ball 3Boj • If instead it is not one of the selected balls, then let Ba. = Ja(Pa., r ao) be the first selected ball that intersects Bo, = B(Po" ro,). Then the radius ra o of BOlo must be at least as great as the radius ro, of B o" otherwise we would have selected Bo, instead of BOlo at the qth step. It follows then from the triangle inequality that 3Ja(Pao ' ra.) == B(Pa., 3rao) covers Ja(Po" ro,). That is what we wished to show. • One of the most important (sublinear) operators in classical analysis is the Hardy-Littlewood maximal operator. We shall apply the Wiener covering lemma to prove the boundedness, in a certain sense, of the Hardy-

86

CHAPTER 3. MEASURES

Littlewood operator. We shall then apply this boundedness to prove the Lebesgue differentiation theorem. De8nition 3.5.2 Let J be a locally integrable function on JlN. We define the Hardy-Littlewood maximal function M J by setting M I(z)

= O
for each x E JRN • It is obvious that the operator M satisfies

What is more useful is the subtle bound, called a weak type (1,1) bound, that we derive in the next lemma. . Lemma 3.5.3 There is a constant C > O-depending on the dimension N, but not on the particular function I -such that the operator M satisfies

lor any scalar A > O. The constant C is independent of A and of I; indeed, C 3N will do.

=

Proof: Fix I E L 1(]RN). Fix A > O. Let s>.

= {x E ]RN : MJ(x) > A}.

We wish to estimate the measure of S>.. By the inner regularity of Lebesgue measure (Proposition 1.3.7), it suffices to measure any compact subset K of S>.. If K is such a compact set, and if k E K, then there is an rio > 0 such that

In other words,

Obviously the balls {B(k,rk)hEK cover K. Thus they satisfy the hypotheses of Lemma 3.5.1. So we may select a subcollection {B(k;.rk;}~l consisting of pairwise disjoint balls such that {B(kj, 3rk; )}~1 covers K.

3.5. COVERING LEMMAS

87

Now we calculate that

~

eN (K)

~

(Q

B(kj, 3rlel

»)

f:e N ( B(kj,3rlel

»)

eN

i=1

= ~3N eN (JB(kj,rk l »)

~

aN ~ Xklel ,r.;) I/(t)1 dt.

But the balls {lR(kj, r"l)} are pairwise disjoint. Thus the last line does not exceed

•

That is the estimate that we wished to prove.

Just to familiarize the reader with a typical, if elementary, application of maximal function estimates, we now prove a version of the Lebesgue differentiation theorem. Proposition 3.5.4 Let I be a locally integrable function on for almost every x E ]RN, it holds that lim r-+O+

]RN.

Then,

1 _. [ I(t)dt eN(B(x,r») iB(z,r)

exists and equals f(x).

Remark 3.5.5 Perhaps a few comments are in order before we proceed with the proof. First, it is common in measure theory books to formulate a stronger version of Lebesgue's result. It is this: Let I be a locally integrable function on ]RN. For almost every x E]RN we let f(x) == lim

r-+O+

eN(~ x, r » iB(z,r) [ I(t) dt,

(3.58)

which limit is guaranteed to exist by Lebesgue's theorem. Then, for almost every x, we have that

lim

r-+O+

eN(~x, r »iB(z,r) [ If(t) -

l(x)1 dt

= O.

(3.59)

CHAPTER 3. MEASURES

88

It is easy to see that statement (3.59) is stronger, on a formalleve1, than

the statement in the proposition (logically speaking, it turns out that the two are equivalent in the sense that .either one implies the other). Notice also that the stronger statement (3.59) contains implicitly an interesting fact. Strictly speaking, an V i1Jnction is an equivolence class of functions-it is only defined up to a set of measure zero. In other words, any given representative of this equivalence class may differ from any other representative on a set of measure zero. However equation (3.58) gives a way to pick a canonical representative from the equivalence class: no matter what the choice of I from the equivalence class, (3.58) gives the same definition of I(z). The set of Lebesgue points-those points at which (3.59) holds-is independent of the representative I with which we work. And the resulting function I(z), defined by (3.58), does not depend .on which representative is used to calculate the limit. . Proof of the Proposition: Let U c JRN be an open set, and let V be an open set that contains U. By Urysohn's Lemma 1.1.3, we may let 4J be a cutoff function that is identically equal to 1 on U and that is supported in V. By multiplying I by o. Choose a continuous function 9 with compact support such that 11/- gll£1 < £2. Set T.

=

{z E

]RN :

I

lim sup .eN r-+O+

-lim inf .eN T-+O+

(lB~z, r »JB(z,r) r I(t) dt

(lB~Z, r » lB(z,r) f I(t) dtl > E}.

We wish to estimate the measure of T •. Now .eN(T.)

~

.eN

+

.eN ({ x E

({x

liminf r-+O+

e]RN :

lim sup r-+O+

.eN(~ Z, r »JB(z,r) r I/(t) -

]RN : llim sup .eN (lB~ r-+O+

z, r

»

.eN(lB~ :1:, r » JB(z,r) r g(t)dtl >

r

JB(z,r) E/

({x e]RN : liminf .eN(~ r » r

+

.eN

-

AI +A2 +A3.

r-+O+

:1:,

g(t)ldt > £/3})

get) dt

3})

JB(z,r)

Ig(t) -/(t)ldt > E/3})

It is plain that A2 = 0 since 9 is continuous. We estimate Al and A3 in

89

3.5. COVERING LEMMAS just the same way, so we shall. concentrate on the former. Now

Al

({z e ~ : ~p £N(~z, r» k."r) e~ > 3})

~

£N

=

£N({z

~

C E/3 111 - gllL1

<

-·E

=

C·E.

C

:M(f -g)

I/(t) - g(t)1 dt

> E/3})

E/

2

E/3

It foll..ows from this last estimate that Al = 0 (if instead Al = ,\ > 0 then take E = '\/(2C) to derive a contradiction). By similar reasoning, As = o. Thus, for E sufficiently small, £N (T.) = o. This gives the result that the desired limit exists. That the limit agrees with the original function (representative) I almost everywhere follows from a similar argument that we leave to the reader. • Corollary 3.5.6 If A C lItN is Lebesgue measurable, then, for almost every

z·e lItN, it holds that . X () .A

Proof: Set

I

x

r

= r.!.If:+

£N(Anlll(z,r» £N(lII(z,r»·

= XA. l'hen (

f(t) dt = &.N(A n lII(x,

JlI(z,r)

r»

and the corollary follows from Proposition 3.5.4..

•

Definition 3.5.7 A function I : lItN -+ lIt is said to be approximately continuous if, for almost every Xo e ]iN and lor each f > 0, the set {x: I/(x) - l(xo)1 > f} has density 0 at Xo, that is,

0- lim .cN({x:l/(x)-/(xo)l>f}nlll(xo,r» - r-+O+ £N(lII(xo,r» . Corollary 3.5.8 If a function it is approximately continuous.

I : ]iN -+

]i

is Lebesgue measurable, then

Proof: Suppose that I is Lebesgue measurable. Let ql, ~, ... be an enumeration of the rational numbers. For each positive integer i, let Ei be the set of points x ~ {z : I(z) < qi} for which

o

lim

<

r-+:!P

.cN({z:/(z)
CHAPTER 3. MEASURES

90 and let Ei be the set of points x ¢ {z : qi O

·

< 1lDlSUP r-+O+

< I(z)} for which

C.N({z:q.
By CoroUary 3.5.6 and the Lebesgue measurability of I, we know that =0 and c.N (Ei) = O. Thus we see that

c,N (&)

00

E

= U(Ei UEi} i=1

is also a set of Lebesgue measure zero. Consider any point Xo ¢ E and any qi and q; such that

I(xo} -

f

f

> O. There exist rational numbers

< qi < I(xo} < q; < I(xo} + f.

We have {x : I/(x} - l(xo}1 > f} C {z : I(z} the definition of Ei and Ej we have

and 0=" lim r-+O+

< qi} U {z : q; < J(z)}. By

c. N ({z : qj < I(z)} n B(xo, r». C,N(B(xo,r»

It follows that

0= lim C,N({x:I/(x}-/(xo}l>f}nB(xo,r»." r-+O+ C,N(B(xo,r» Since Xo ¢ E and f > 0 were arbitrary, we conclude that J is approximately continuous. • Remark 3.5.9 The converses of Corollaries 3.5.6 and 3.5.8 are also true, but there is more technical difficulty involved when one deals with a set A or a function J that may not be measurable. The interested reader should see Federer [41, §2.9.12 and §2.9.13. The Besicovitch Covering Theorem We next turn to Besicovitch's covering theorem because its proof is similar to that of Wiener's theorem. It is a purely geometric lemma, as it speaks to how balls may be packed in space.

3.5. COVERING LEMMAS

91

Theorem 3.5.10 (Besicovitch) Let N be a positive integer. There is a constant K = K(N) wi~.~ following property. Let 8 = {Bj}~l be any finite collection 0/ open balls in RN with the property that no ball contains· the center 0/ any other. 77len we may write

so that each 8j, j

= 1, ... ,K, is a collection o/pairwise disjoint balls.

In general it is not known what the best possible K is for any given dimension N. Interesting progress on this problem has been made in Sullivan [1]. Certainly our proof will give little indication of the best K. We shall see that the heart of this theorem is the following lemma about balls. We shall give two different proofs of this lemma. One, contrary to our avowed philosophy in the present subsection, will in fact depend on measure-or at least on the notion of volume. The second proof will rely instead on trigonometry. . Lemma 3.5.11 There is a constant K = K(N), depending only on the dimension 0/ our space aN, with the following property: Let Bo = B(xo, TO) be a ball 0/ fixed radius. Let Bl ~ B(Xl,·Tl),B2 = B(X2,T2), .•. ,Bp = B(xp, Tp) be balls such that

(i) Each B j has non-empty inters·ection with B o, j (ii) The radii Tj

~

= 1, ... ,Pi

TO lor all j = 1, ... ,Pi

(iii) No ball Bj contains the center 0/ any other B" lor k

¥-

j, j,k

=

O, .•• ,p.

Thenp

~

K.

Here is what the lemma says in simple terms: fix the ball Bo. Then at most K pairwise disjoint balls of (at least) the same size can touch Bo. Note here that being 'pairwise disjoint' and 'intersecting but not containing the center of any other ball' are essentially equivalent: if the balls intersect, but none contains the center of another, then shrinking each ball by a factor of one half makes the balls pairwise disjoint; if the balls are already pairwise disjoint, have equal radii, and are close together, then doubling their size arranges for them to intersect without any ball containing the center of another. First Proof of the Lemma: The purpose of providing this particular proof, even though it relies on the concept of volume, is that it is quick and intuitive. The second proof is less intuitive, but it introduces the important idea of 'directionally limited.' Since any ball Bit j = 1,2, ... ,p, for which Tj > TO can be replaced by the ball of radius TO internally tangent to B j at the point of intersection

CHAPTER 3. MEASURES

:u.

Figure 3.4: Replacing a Larger BaIl of the segment connecting the center of Bj and Bo while maintaining the validity of all three conditions (i)-(iii) (see Figure 3.4), it will suffice to prove the lemma with all balls having radius roo We now assume that all balls have the same radius. With the balls as given, ·the smaller balls Ja(Xl' ro/2), Ja(X2' ro/2), ... , JR(xp, ro/2) are pairwise diSjoint and are all contained in JR(xo,3ro). We calculate that p

eN(u'J=lBj) (ro/2)NYN eN (JR(xo, 3ro» (ro/2)NYN

As a result of this calculation, we see that K(N) exists and does not exceed 6N • • Second Proof of the Lemma: For this argument, see Krantz and Parsons [1]. In fact we shall prove the following more technical statement

Let the universe be the two dimensional plane, ]R2, and let ~ = {reill : 0 $ r < 00,0::; (}::; 1r/6}. Set S = {z E ~: Izl ~ 3}. H a,b E S and if each of the balls lR(a, r), lR(b, s) intersects 111(0, 1), then

la - bl < max(r, 8).

(3.60)

93

3.5. COVERING LEMMAS

A moment's thought reveals that this yields the desired sparseness condition in dimension two. The N-dimensional result is obtained by slicing with two dimensional planes. To prove (3.60), we first note the inequalities (0

_1)2 - (2 - va)a2 ~ 0

if 0

~

({3 - 1)2 - (02 - ..r3a{3 + {32) ~ 0

3;

(3.61) (3.62)

The first of these is proved by noting that the derivative of the left side of (3.61), in the variable 0, is positive when 0 ~ 3; and the inequality is satisfied when 0 = 3. So the result follows from the fundamental theorem of calculus. Similarly, the derivative of the left side of (3.62), in the variable {3, is positive when {3 ~ 0 ~ 3, and the case {3 a 3 is just inequality (3.61), which has already been established. With these inequalities in hand, we introduce polar coordinates in the plane, writing a = oei8 and b = {3e i . We assume without loss of generality that 0 ~ {3. The hypothesis that B(b, s) n B(O, 1) '# 0 entails s > f3 - 1; thus it suffices to show that

= =

(3.63) The law of cosines tells us that

la - bl 2 = 0:2 -

2a{3cos(¢> - 9)

+ (P.

(3.64)

Since cos(¢> - 9) ~ cos7r/6 = .../3/2, it follows that the right side of (3.64) does not exceed a2 - v'3ab + b2 • Tlie inequality (3.63) now follows from • (3.62). H. Federer's concept of a directionally limited metric space-see Federer [4]-formalizes the geometry that goes into the proof of our last lemma. More precisely, it generalizes to abstract contexts the notion that a cone in a given direction can contain only a certain number of points with distance T/ > 0 from the vertex and distance T/ from each other. The interested reader is advised to study that primary source. Now we can present the proof of Besicovitch's covering lemma: Proof of Theorem 3.5.10: Begin as in the proof of the Wiener covering lemma. Select BI to be a ball of maximum radius. Then select B~ to be a ball of maximum radius that is disjoint from Bl. Continue until this selection procedure is no longer possible (remember that there are only finitely many balls in total). Set B1 {BJ}. Now work with the remaining balls. Let B? be the ball with greatest ra.dius. Then select B~ to be the remaining ball with greatest radius, disjoint

=

94

CHAPTER 3. MEASURES

from

B2

B1.

Continue in this fashion until no further selection is possible. Set

= {Bj}.

Working with the remaining balls, we now produce the family Ba, and so forth. Clearly, since in total there are only finitely many balls, this procedure must stop. We will have produced finitely many-say p-nonempty families of pairwise disjoint balls, Bl"'" 8". It remains to say how large p can be. Suppose that p > K(N) + I, where K(N) is as in the lemma. Let Bf be the first ball in the family B". That ball must have intersected a ball in each of the preceding families; by our selection procedure, each of those balls must have been at least as large in radius as Bf. Thus Bf is an open ball with at least K(N) + 1 "neighbors" as in the lemma. But the lemma says that a ball can only have K(N) neighbors. That is a contradiction. • We conclude that p $ K(N) + 1. That proves the theorem. Using the Besicovitcb covering theorem, we now can extend the earlier results on the maximal function and differentiation to Radon measures other than the Lebesgue measure. Definition 3.5.12 Let p be a Radon measure on ]RN and let I be a locally p-integrable function on ]RN. We define the" Inaximal function with respect to p, denoted M,./, by setting M,./(x) = 0 if there exists 0 < r < 00 with p(lII(x, = 0, and

r»

M,.I(x)

1

=O
B(."r)

lI(t)1 dpt

otherwise." Lemma 3.5.13 There is a constant C > O-depending on the dimensilm N and the Radon measure 1', but not on the particular function I -such

that the operator M satisfies p{x

N

E]R

:

M,.I(x) > A} $

C -:xIl/IlL!(RN)

lor any scalar A > O. The constant C is independent 01 A and 01 I. Proof: The same proof may be used for Lemma 3.5.13 as was used for Lemma 3.5.3, except that the Besicovitch covering theorem replaces the Wiener covering lemma and a little more care must be exercised in constructing the initial cover of K so as to arrange that no ball contain the center of any other ball. • As a corollary, we get the following more general differentiation theorem: Proposition 3.5.14 Let p be a Radon measure on]RN and let I be a locally IJ-integrable function on ]RN. Then for p-almost every x E ]RN it holds that lim r-+O+

1 p(B(x,

r»

( JIl(."r)

I(t) dpt

3.5. COVERJNG LEMMAS

95

exists and equals /(x). Lebesgue's Covering Theorem The covering theorems presented thus far can be adapted to a variety of situations (besides Euclidean space); notable among these is the setting of spaces of homogeneous type (see Christ [1], Coifman and Weiss [1], [2], Krantz [5]). The next theorem is an integral part of dimension theory (for which see Hurewicz and Wallman [I]), which is usually studied on a separable metric space. The theorem has important geometric content, and is of wide utility. Note that if U = {U.. } ..EA is a covering of a set S by open sets then a refinement V = {V.s}.sEB of U is an open covering of S with the property that each V.s lies in some U... Theorem 3.5.15 Let S c JRN be any set. LetU = {U.. } .. EA be a covering 0/ S by open sets. Then there is a refinement V = {V.s}.sEB 0/ U that has valence at most N + 1. That is, each point of S is an element 0/ at most N + 1 0/ the V.s. The proof will proceed in several steps. We will introduce a few simple concepts from dimension theory. First, the empty set is declared to have dimension "-1. It is the only set with dimension -1. A point P in a set S is said to have dimension $ N if P has a neighborhood basis {Uj } with the property that aUj n S has dimension $ (N -1). A set is said to have dimension N if it has dimension $ N but not $ (N - 1). A moment's thought reveals that.th~ set of rational numbers, lying in lR, has dimension O. So does the set of irrational numbers. For the first of these, simply note that if P is rational then a ball around P with irrational radius will have boundary possessing intersection with Q that is empty (hence has dimension -1). It will be useful for us to note that if 1 $ m $ N then the set T m of points (Xli X2, ••• , XN) in l\lN with Xli X2, • •• , Xm rational has dimension O. Thus we see that

where S is the set of points in JRN with all irrational coordinates. And each of these sets has dimension zero. Thus we have decomposed JRN into (N + 1) sets of dimension O. While it is certainly true that a discrete set is of dimension 0, the preceding examples show that the converse is not true. Nevertheless, sets of dimension 0 share certain properties with discrete sets. The one of greatest interest for us is contained in the following lemmas. Before beginning, we note that a set C C JRN is said to separate two disjoint subsets AI, A2 of RN if JRN \ C = UI U U2 with UloU2 being relatively open in JRN \ C,

CHAPTER 3. MEASURES

96

disjoint, and Aj C Uj for j = 1,2. As an exercise, the reader may wish to verify that if a set S C lllN is zero dimensional then any two disjoint relatively closed sets in S may be separated by some set 0 c S (or see stateIqent (E), page 15, in Hurewicz and Wallmah [1]). Lemma 3.5.16 1/ E1> ~ are closed, disjoint subsets 0/ RN and if B is a zero dimensional subset 0/ aN, then there is a closed set C in aN such that (i) C separates El and ~ and (ii) B n C = 0. Proof: There certainly exist open sets WI, W2 such that Ej C Wj, j = 1,2, and WIn W 2 = 0. [This is just the property of normality for JRN.J Now WIn B and W 2 n B are disjoint and closed in B. Since B has dimension 0, these two sets can be separated in B (by the remark preceding the statement of this lemma). We conclude that there are disjoint sets U{, U~ satisfying B=U;UU~

with, W j nBc Ui, j =.1,2, and U{, U~ both closed and open in B: Now (U; nW 2 )u (U~

n WI)

= 0,

(3.65)

(Ui n U'2) u (U~ n U'l) = 0,

(3.66)

and therefore (U;

nE2 ) u (U~ n Ed = 0.

(3.67)

But WlJ W2 are open, so that (3.65) implies that (U'l n W 2 ) U (U'e- n = 0. As a result, by the first sentence of the proof, we conclude that (U'l n~) u (U'2 n E l ) = 0. Equations (3.65}-(3.67) allow us now to conclude, since El n ~ = 0, that neither of the disjoint sets El U U{ and E2 U U~ contains a cluster point of the other. Since every subspace of lllN is normal, we find that there is an open set 0 such that

Wd

and on(~ U U~)

In conclusion, the boundary C disjoint from U{ U U~ = B.

=0

= 0.

\ 0 separates El and

~

and is •

Lemma 3.5.17 Let S be a zero dimensional subset o/JRN . Let U = {Ul, U2} be open subsets o/lllN that cover S. Then there is a refinement V = {Vt, V2} ofU that still collers S and such that Vt n V2 = 0.

3.5. COVERING LEMMAS

97

Proof: It is convenient to replace the universe RN with the set X = U1 UU2 and we do so without further comment. Thus D1 = X \ U2 and D2 = X \ U1 are closed, disjoint sets. Since S has dimension 0, we may apply the preceding lemma to get a closed set C that is disjoint from S and separates D1 from D 2 • Thus there exist sets Wh W2 which are open, disjoint, and satisfy Wj ::> Djo j = 1,2, and X\C = W1 UW2 • As a consequence, Wj C Uj, j = 1,2. Also SnC = 0 and W1 UW2 ::> S. This is the desired conclusion. • LeDlDla 3.5.18 Let S be a zero dimensional "Subset of aN. Let U = {Uj}~=r1" be open sets that colier S. Then there exists a refinement V = {V; }~~1 of U that still covers S and such that V; n V" = 0 when j :f: k. Proof: We proceed by induction on p. Note that if p = 1 then the statement is trivial. Now assume that the result has been established for p = k - 1; we use that hypothesis to prove the result for p = k. Thus we are given a covering U =" {Uj of a set sCaN. Let Uk_1 == U"-l U U". Then the collection U' given by

H=l

is still an open covering of S; of course it has (k-l) elements. The inductive" hypothesis applies to the covering U' of S, and we obtain a refinement

of U' that still covers S and is such that the elements of V' are pairwise disjoint. Observe that the set T == snvk_1 has dimension not exceeding 0 (since S does). Notice also that U"-l and U" cover T. By the last lemma, there exist open sets V"-l> V" that refine the cover U,,-l> U" of T and which are disjoint. But then form an open covering of S with all the desired properties.

•

The proof of the Lebesgue covering theorem is now easy: Proof of Theorem 3.5.15: Begin by writing S = Sl US2 U···USN+1,

where each Sj has dimension at most O. Since our open covering certainly covers each Sj, we may apply the last lemma to obtain (N + 1) refinements

CHAPTER 3. MEASURES

98 VI, ... , VN+l of U such that

Vi = {Vii, ... , V;(i)};

(3.68)

Vi covers Si;

(3.69)

V; n Vt' = 0

if 8

I t.

(3.70)

Vl

Now let V be the covering consisting of all the open sets for all possible values of i and j. This covering certainly covers S, for it covers each of the Si. We claim that V has valence at most N + 1. For note that any selection of N + 2 elements of V must contain two elements from a single Vi which are therefore disjoint. So it is impossible for N + 2 elements of V to have a point in common. • It turns out that the property of any open covering having a refinement of valence at most N + 1 characterizes sets of dimension N. Since this is not a study of dimension theory, it would take us far afield to provide all the details of this assertion. We refer the interested reader to Hurewicz and Wallman [1] .

. The Vitali Covering Theorem The Vitali covering theorem is probably the most broadly used covering theorem in all of measure theory. It has the dual advantage of being both intuitively appealing and rather profound. Indeed it says something very important about the structure of sets in Euclidean space. We derive our treatment of Vitali's ideas from Evans and Gariepy [l]i the interested reader may consult that source for further details. We begin with a modification of the Wiener covering theorem: Proposition 3.5.19 Let U be a collection of closed balls in RN such that

sup{diamB: B E U} = M Then there is a countable, disjoint subfamily V

UBe BeU

< 00.

eU

such that

USB. BeV

[Here, as usual the notation 5B denotes the ball with the same center as B but 5 times the radius.] The proof of this proposition is left to the reader. Now the principal theorem of Vitali type that we wish to consider is this:

3.5. COVERING LEMMAS

99

Theorem 3.5.20 (Vitali) Let U c aN be an open set. Let 6 > O. Then there e:£ists a countable collection W = {Bj}~l 0/ disjoint closed balls, each lying in U, such that . ". .

(i) diamBj (li)

~

6/or each j;

CN[U\U~l~;] =0.

The gist of the theorem is that one may "fill up" an arbitrary open set U with small balls, to the extent that the portion not covered has measure zero. Of course it is too much to hope that the entire open set can be exhausted by balls. For example, if U is the interior of a unit square in the plane, then it is easy to see that there is no way to fill U completely with countably many discs of radius not exceeding 1/4. In fact, one can see in this example that the set of points not covered will be uncountable and nowhere dense. Proof: Assume for convenience that U has finite Lebesgue measure. Some remarks about the general case will be given at the end. Fix a number A such that 1 - 1/5N < A < 1, where N is the dimension. . We will construct the required baIls by an iterative procedure. Each step of our process will exhaust a proportion (1 - 9) of that portion of U that is not already covered by balls. For the first step, set U1 = U and let

(h = {B : B is a closed ball lying in Ul and having diameter less than 6}. By the preceding proposition, there iLa countable, disjoint family 'HI C 91 'with the property that Ul C 5B.

U

BE1l1

As a result, we may calculate that CN(Ul)

~

L

£N(5B)

BE1l1

=

5N

L

£N(B)

BE1l1

=

5N

£NCU

B).

BE1l1

We conclude that CN(_U B) BE1l1

~ 5~£N(Ul).

100

CHAPTER 3. MEASURES

Since each element of 1i1 is a subset of Vb we see that

Of course the collection of balls 1i 1 is countable; thus there is a subcollection B I .B 2 , •••• BK, satisfying f:.N (VI \

0

Bi )

~ Am(V.).

)=1

For the second step. set K,

V2

= VI \

U Bi i=1

and let

{h

= {B : B is a closed ball lying in V2 and having diameter less than 6}.

Repeating the arguments from Step 1, we find a collection

BK,+l,

BK, +2,

... , B K, of disjoint elements of ge such that

=

f:.N (V2 \

U i) B

j=M,+1

~ ~

Ae N (V2 ) >.2 f:.N (U).

We continue this process countably many times. The result is a pairwise disjoint collection of closed balls B j, all lying in U, all having diameter less than fJ, such that f:.N (V

\Q Bj) ~

>.p eN (V),

for p = 1.2, .... Since 0 < A < 1, our conclusion follows provided that f:.N (U) < 00. In case f:.N (U) = 00. write V = (UlUl) U T, where each Ut is open with finite measure and T has measure zero. •

3.6

Functions of Bounded Variation

Functions of bounded variation from R to IR are well understood (see Federer (4], §§2.5.16-2.5.18). The situation for functions from aN to R is

101

3.6. FUNCTIONS OF BOUNDED VARIATION

less settled and there would seem to be opportunities for future clarification. Our main references for this section are Evans and Gariepy [1] and Ziemer [1]. The following example motivates the definition of functions of bounded variation.

Example 3.6.1 Suppose I E C'2(n), where n is an open subset of IRN . For any 9 = (gl.!h, ... ,gN) E c:(n;IRN ), with Igi ~ 1 for all x E n, we can use integration by parts to see that

L

Idiv(g)dx

=-

L

(grad/)· gdx

~

LI

grad II dx.

=

Now, set n' n n {x : grad/(x) ¢ O}. If we let K be an arbitrary compact subset of n', then we can choose a non-negative Coo function ¢J with 0 ~ t/J ~ 1, suppl en', and with tf>(x) 1 for x E K. Setting

=

9

= -¢Jgrad f

Igrad 11- 1 ,

we have 9 E cJ(n; KN), with Ig(x)1 ~ 1 for all x En, and in this case, we have

L

I div(g) dx

=

L

tf> Igrad II dx

~

L

Igrad II dx.

Since Ken' was an arbitrary compact set, we conclude that

L

Igrad II dx SU P

=

{L /

19(x)l~IVXEn}.

N diV(9)dx: 9 EC:cn;IR ),

(3.71)

The reader should note that with more effort one can see that (3.71) also holds for / E c 1 (n). Definition 3.6.2 Let n C IRN be open. A function be 01 bounded variation on i/

n

I

E Ll (n) is said to

LID/I:= sup

{fo

Idiv(g)dx: 9 E c:
The space 0/ functions and will be normed by

),

Ig(x)l

0/ bounded variation on

II/liB\'

= II/IILI +

L

~ 1 "Ix En}

<

00.

(3.72)

n will be denoted BV(n)

ID/I·

(3.73)

Basic, but important, properties of BV functions follow easily from the definition.

102

CHAPTER 3. MEASURES

Proposition 3.6.3

III e BV(U) and 9 e

: IL

ldiv(g) dzl

C~(U;lRN), then

~~plgl

k

(3.74)

IDFI .

•

Proof: Immediate from the definition. Proposition 3.6.4 III e BV(U) and u both bounded, then ul e BV(U) with

e

a1 (n)

with lui and Igrad ul

f ID(u/)ISsuplul f IDII+suplgradul f I/ldz. 10 0 10 0 10

(3.75)

Proof: Use the identity udiv(g)

= div(ug) -

(grad u· g)

•

and (3.72).

Theorem 3.6.5 (Semicontinuity) Suppose U C RN is open. 11ft, 12, ... is a sequence in BV(U) that converges in Ltoc(U) to a function I, then

f IDIsI. 10f ID/I S li~inf .o~oo 10

(3.76)

Proof: Let 9 E a~ (Uj RN) with I!!I S 1 be arbitrary. Then for any i we have .lim f Ii div(g)dz S l~infIDIsI. 10f /div(g)dz = 0-+00 10 o~oo The result follows by taking the supremum over all such g. • Corollary 3.6.6 (Completeness) The space 01 functions 01 bounded variation with the nonn given in (3.73) is a Banach space. Proof: The reader may verify that (3.73) does in fact define a norm. We will verify completeness. Suppose 11,12, ... is a Cauchy sequence in the norm II·IIBV. A lortiori, 11,12, ... is a Cauchy sequence in the Ll norm, and thus converges in the Ll norm to a function I e LI(n). We know that Is -Ii converges to Is - / in Ll(U) as i -+ 00, so by the Semicontinuity Theorem 3.6.5 we have lim sup

i~oo

f

10

ID(1s - 1)1 S S

lim sup (liptinf ( ID(fi i~oo 3~OO 10 limsup

i.i~oo

f

10

ID(fi -

1;)1 = 0,

1;)1)

3.6. FUNCTIONS OF BOUNDED VARIATION

103

•

from which the result follows.

The next theorem shows that BV functions can also be characterized as those functions whose partial derivatives are measures. TheorelD 3.6.7 Let 0 c JlN be open. 1// E BV(O) and 1 ~ i ~ N, then there e:dsts a signed Borel measure I'i 0/ finite total variation on 0 such that (3.77)

lor each tP E C~(O). Conversely, i/ there are signed Borel meosure8 1'10 1'2, ... ,I'N such that (3.77) holds/or each 1 ~ i ~ Nand tP E ~(O), then / E "BV(O). Remark 3.6.8" Note that the left-hand side of (3.77) represents the action on the test function tP of the partial derivative of / in the sense of distributions. Proof: Suppose that / E BV(O). The left-hand side of (3.77) defines a linear operator Li on C~(O). By (3.74), we see that L is bounded in the supremum norm. The lliesz representation theorem (see Rudin [2], Theorem 6.19) then insures the existence of a signed measure such that (3.78)

tP E C~(O). (Note that since Li is bounded in the supremum norm, it ~en~~~~J " To see the converse, simply observe that for (0; "liN ) with Igl ~ 1,

for

gee:

we have

1

/div(g)dx

{}

N

=- L i=1

L -

gidpi $

{}

N

L Il'il(O). i=1

•

Definition 3.6.9 II IE BV(O), then DI is used to denote the derivative in the sense 01 distributions. Corollary 3.6.10 II/ E BV(O), then there exists a Radon measure I' on

o and a I'-measurable function u, with lu(x)1 = l/or I' almost every x E 0, such that D/=up.

An important point to make in reference to Theorem 3.6.7 is that the measures representing the partial derivatives of a function of bounded variation typically have a non-trivial singular part with respect to Lebesgue measure. Whenever the singular part with respect ~ Lebesgue measure is non-zero, then the function in question cannot be in any Sobolev space (Sobolev spaces will be treated in some detail in the next chapter). The next example illustrates this phenomenon.

104

CHAPTER 3. MEASURES

Example 3.6.11 Set

J = Xi(O.I)' the characteristic function of the closed unit ball in JltN. By the divergence theorem2 , for 9 e C: (aN , R,N) we have

{ li(o.l)

div(g)dx =

(

g. vd'H. N - 1 ,

ls(o.l)

where v is the outward unit normal to the unit sphere. Comparing this equation with (3.77), we ~ that DJ = _v1l N - 1 LS(O, 1), the measure 1l N - 1 LS(O, 1) is singular with respect to Lebesgue measure. The closure of the space of Coo functions in the BV norm is the Sobolev space ·Hl.l of functions whose partial derivatives in the sense of distributions are represented by Ll functions. Thus in general it is impossible to approximate a BV function in the BV norm by a smooth function: On the other hand, the next theorem gives us some positive information on smooth approximations to BV functions. Theorem 3.6.12 Let J e BV({}). Then there exists a sequence of functions It, 12, ... in Coo({}) such that

(ii)

In IDhl-t In IDfl,

(iii) Dh -" Df Proof: Let '1/;" be a family of mollifiers as in Section 1.3. While it is not essential, it is convenient to suppose that '1/;" satisfies the symmetry (evenness) condition (3.79) ¢,(-x) = 'I/;,,(x). In any case, one can always arrange for (3.79) to hold by replacing 'I/;,,(x) by H¢,,(x) + '1/;,,( -x». Technical difficulties are minimized if we assume that {} = JltN, so we shall first make that assumption. 2We use the tenns Gauss-Green theorem and divergence theorem interchangeably. Both are multidimensional versions of the Fundamental Theorem of Calculus (see Section 5.5)

3.6. FUNCTIONS OF BOUNDED VARIATION

105

Fix I E BV(IllN). By Theorem 1.3.31, we know that 1* tPa -+ I in Ll(IllN), proving (i). Suppose 9 E C~(RN;aN). Then using Fubini's Theorem and the theorem on differentiating under the integral sign, compute

we

I (J * tPa)(z) div(g)(z) d:i: JRN = kN (kNI(Y)tPa(Z-Y)dY)

div(g)(z)dx

=

kNI(Y) (kN tPa(Z-Y)diV(9)(Z)dx) dy

=

kNI(Y) (kNtPa(-Z)diV(9)(Y-Z)dx) dy

=

kNI(Y)diV (kN 1Pa(-z)g(y-z)dx) dy

=

I I(Y) div(g* 1Pa)(y)dy; 1RN

conclusion (iii) now follows, because Theorem 1.3.31 tells us that g*tPa converges to 9 in the supremum norm as (J' -+ O. Also note that I(g * tPa) (z) I $ 1 . holds for all z, showing that

But of course Theorem 3.6.5 implies that

so (ii) also follows. To handle the general case, that I E BV(O) when 0 =F lllN, we must show how to construct a function U E COO(O) with

where e > 0 is given. To this end, choose an open 0 0 CC 0 with

I

10\00

ID/I $

-3t

and construct a sequence of open subsets of 0 with 0 0 CC 0 1 CC ... C 0 and U~OOi = O. Next, introduce a partition of unity .pi, i = 0, 1, ... , such that

1/>0

= 1 on 00,

supp (1/>0) C 010

106

CHAPTER 3. MEASURES

and in general, supp (tPi) c niH \Oi-1o i ~ 1. Of course, we have L:o tPi == Ion n. For each i = 0, 1, ... , we choose tTi > 0 so that with

=

Ui (ftPi) * ""tTl'· the fOllowing conditions hold: supp (Ui)

L

L

= (J gradtPi) * tPtTl'

supp (Uo) c nlo niH \ Oi-l, i ~ 1,

(3.80) (3.81)

ItPil clz < 2-(iH) ~,

(3.82)

< 2-(iH) ~.

(3.83)

c

lUi -

Vi

IVi - fgradtPil clz 00

u= LUi, i=O

and notice that U E COO(n) because each point of n is only in the support of finitely many Ui. The estimate of the Ll norm of the difference between f and U is simply

{ Iu - II clz ~

10

f: 10( IUtPi) * .,pa, - frPildz
The more interesting estimate bounds the quantity 10 IDul." Using the fact that 00

o = grad(E:o tPi) =

L grad 4>i, i=O

we compute Du

= =

00

00

i=O

i=O

:Eu grad tPi) * ""a, + L(4)i D f) * tPa, 00

00

i=O

i=O

:E[U grad tPi) * .,pa, - f grad4>d + L(4)i D f) * .,pa,.

Arguing with Fubini's Theorem as in the case n {1(tPiDf)*""a,IS{

10

_

=]RN, we see that

101+1 \0'_1

IDflj

since each x E n belongs to at most three of the sets able to estimate

ni+l \ Oi-1o we are

LIDul ~

•

107

3.7. DOMAINS WITH FINITE PERIMETER

The final result we will present in this section concerns compactness in the space of functions of bounded variation. -The compactness result is a consequence of the smooth approximation theorem just proved and Rellich's theorem which we will state without proof. A general treatment of Rellich's Theorem can be found in Ziemer [IJ, Section 2.5. Decreasing the regularity of 80 increases the difficulty of proving the result. The relevant version of the theorem with nearly the weakest possible regularity' assumption for 80 is the following: Theorem 3.0.13 (Rellich) Suppose n c lllN is a bounded open set the boundar7J of which can locally be represented as the graph of a Lipschitz function. If h, /2, ... is a sequence of functions in Ll (0) whose first partial derivatives in the sense of distributions are also in Ll(O), and if there is M < 00 such that

f

leI"

Ifil dx +

10f IDII dx ~ M

for i = 1,2, ... ,

then a subsequ~nce h; converges to a function f E Ll(O). Corollary 3.6.14 (Compactness) If O.C lllN is a bounded open set the boundar7J of which can locally be represented as the graph 0/ a Lipschitz function, then any set 0/ functions bounded in· the BV nonn has a subsequence that converges in Ll(O) to a function sn BV(O). . Proof: By the Approximation Theorem 3.6.12 we can choose functions hI. h 2 , ••• in COCCO) with

10f IIi -

hil dx

~~

and

z.

10f IDhil ~ M + 1.

Then Rellich's Theorem allows us to extract a subsequence hi; that converges in the Ll norm to fELl (0). A fortiori we know that h; converges to / in the Ll norm and we have f E BV(O) by the Semicontinuity Theorem 3.6.5.

•

3.7

Domains with Finite Perimeter

Definition 3.7.1 (i) If S is a Borel set and 0 C lllN is open, then the perimeter of S in o is denoted by peS, 0) and is defined by P(S, 0)

=

L

=

Sup{!s div(g) dCN : 9 E C:(O; lllN),

IDxsl Igl

~ I}, (3.84)

108

CHAPTER 3. MEASURES where Xs denotes the chamcteristic function

(ii) We say

S

is

01 locally finite

0/ S.

perimeter if

P(S,O) < 00 holds lor every bounded open set O. Sets also called Caccioppoli sets.

(3.85)

0/ locally finite perimeter are

(iii) If S is 01 locally finite perimeter, then there is a positive Radon measure p. and a p.-measumble]RN -valued function u, with lu(x)1 = 1 lor p.-almost every x, such that DXs = up. (see Corollary 3.6.10). It is customary to use the notation 118S11 for the Radon measure p. and to write· Vs lor the function u, so that (3.86)

Dxs = Vs 118811·

(iv) We write peS) = pes, JRN), and we say 8 is o/finite perimeter if P(8).< 00. Example 30702 II S is a C l dqmain, then 8 is of locally finite perimeter. To see this, suppose p is a Cl defining function for 8 so that

< O}

S = {x : p(x)

and Vp(x) =1= 0 for x E 8S. Let ifJ : JR -+ JR be any Coo function with ¢'(t) :5 0 for all t and ifJ(t) =

{Io

~ft:5

-1,

If 0:5 t.

Fixing a bounded open set 0 and a function 9 E C~(n;JRN) with we use integration by parts to estimate

is

div(g) deN

=

Igl

:5 1,

JRN ¢(p/t) div(g) d£N

lim ( t.j.o

JRN V [ifJ(p/t)] 0gd£N

-lim ( t.j.O

=

-lime l t.j.O

~

{ ¢'(p/t) (Vp) ogdCN JRN

- (inf ifJ') lim e t.j.O

l

{

IVpi dCN .

Jnn{IZ:-t$p(IZ)$O}

The last integral is finite because n is bounded and Vp is non-vanishing on 88.

3.7. DOMAINS WITH FINITE PERIMETER

109

In fact, the term (inf t/J') in the preceding argument can be made as close to -1 as we wish. Also, careful use of the change of variables formula for integration allows one to see that

lim t- 1 t.J,O

IVpi d.eN = 1lN-1 (0 n 8S).

{

Jnn{z:-tSp(z)SO}

We conclude that, for a Cl domain S, P(S,O) holds for each bounded open set

= 1lN - 1 (n n 8S)

(3.87)

n.

The following lemma is an immediate consequence of the fact that the integral of the divergence of a function with compact support is zero. Lemma 3.7.3 Suppose S has locally finite perimeter and open set. Then

118(JRN "and

\

S)II

n is a bounded

= 118SII,

pen \ s, n) = pes, n)

hold.

Proof: Note that, for 9 E C: (:JRN; ]RN), we have { div(g).e N = - (

15

div(g) .eN.

JRN\S

•

By Theorem 3.6.7, if S C ]RN "is a domain with finite perimeter, then the following formula holds for any 9 E (JRN ; JRN) :

C:

Is

div(g)d.e N = -

J

g·vsdIl8SII·

(3.88)

Equation (3.88) can be considered to be an abstract form of the classical Gauss-Green formula (or synonymously the divergence theorem) which we will discuss in more detail in Section 5.5. One aim of the study of sets of finite perimeter is to improve our understanding of the right-hand side of (3.88). As a first step, the next lemma extends the applicability of (3.88). Lemma 3.7.4 Suppose S C ]RN is of finite perimeter. If 9 E ~(JRN ; JRN) is differentiable .eN -almost everywhere and {

JRN then (9.88) holds for g.

Idiv(g)1 d.e N <

00,

(3.89)

CHAPTER 3. MEASURES

110

Proof: Let 1/JtT, 0' > 0, be a family of mollifiers as in Section 1.3. Then (3.88) applies to 1/JtT * g, so that

Is

tPtT*(divg)d£N

=

Is

div(tPtT*g)d.CN =-- !(t/J,.*9)'"sdIl 8SI1 (3.90)

Is

holds. As 0' .J.. 0 the left-hand side of (3.90) coriverges to div(g) deN and the right-hand side converges to I g'''sdIl8SII, both by Theorem 1.3.31. I

Remark 3.7.5 The preceding lemma admits the use of vector fields that are piecewise continuously differentiable and of those that have singular differentials on small sets in the Gauss-Green formula. The next lemma contains a particular instance of the Gauss-Green formula that will be useful later in this section. Lemma 3.7.6 Suppose S C JRN is of 10000ly finite perimeter and 9 E C:(JRN; JRN). For x E JRN, {

lsrot(z,r)

div(g)d.c N

={

g"lIsdIl8SII+ (

lsnsc2l,r)

llf(z,r)

g'I:=:I,crnN-1y, (3.91)

holds for £l-almost every r > O. Proof: Fix r > O. For

E

> 0, define g(y)

r+E-IY-2I1 ()

g.(.) - {

•

Iy - xl :5 r, if r :5 Iy - xl :5 r + E, if r + E :5IY - xl. if

9 Y

o We compute divg(y)

o

Iy - xl ::5 r, if r :5 Iy - xl ::5 r + E, if r + E::5 Iy - xl. if

2I 1dl"V ( ) _ ..Tl=.!.-. ( ) div .. (y) - { r+'-I,I• 9 Y 'llI-%1 9 Y By Lemma 3.7.4 we have

Is

div g. d.c N

=/

g. "115 d1l8SII·

Assume r is such that S n Sex, r) is tiN-I-measurable, which is true for .cl-almost every r > O. Then the result follows by letting E decrease to

O.

I

Area Minimization and the Isoperimetric Inequality Another reason for our interest in sets of finite perimeter is that they provide a context in which to study problems of area minimization.

3.7. DOMAINS WITH FINITE PERIMETER

111

Theorem 3.7.7 (Existence of Area Minimizers) Let U c JRN be a bounded open set, and let B be a set 0/ finite perimeter. Then there e:&ists a set S 0/ finite perimeter with S \ B \ U such that

u=

peS)

S P(T) lor any T with T \ U

= B \ U.

Proof: Let SI, S2, •.. be a sequence of sets of finite perimeter with S. \ U = B \ U, for i = 1,2, ... , and

.lim

• -too

P(S.)

= inf{P(T) : T \ U = B \ U} .

Let K be a compact set that contains U. Using the Compactness Theorem 3.6.14 to pass to a subsequence, but without changing notation, we may suppose that Xs,nK converges in £1 (K) to a function /. Passing to another subsequence if necessary, we can suppose that Xs,nK also converges almost everywhere to I, thus I is the characteristic function of a set C. We set S= CU(B\K), and the result follows from the Semicontinuity Theorem 3.6.5.

•

One of the central facts of classical Euclidean geometry is the Isoperimetric Inequality. A version of this result can be easily proved in the present context of geometric analysis. Later, in Sections 7.2 and 7:3, we will approach the Isoperimetric Inequality using Steiner symmetrization For this purpose we need a simple Sobolev type lemma. The general thrust of a "Soholev Inequality" is to use an integral norm of a higher order derivative to bound an integral norm of a lower order derivative. Actually, this was first done by Poincare on a bounded domain for the L2 norm of the first derivative of a function and the £~.Jlorm of the function, so that particular type inequality is often called Poincare's Inequality. Lemma 3.7.8 If / E BV(JRN

(klV III b

),

then the following inequality is valid:

lV-I deN) r r

S klV IV fl d£N.

(3.92)

Remark 3.7.9 This result can be generalized to other exponents and applies to functions in the appropriate function space, namely Soholev space (see Evans and Gariepy [1]). In the next chapter, we will give the formal definition of Soholev space and present the main theorems concerning such spaces.

=

Proof: We will give the proof in the case N 3. This case illustrates all the salient points without excessive notational complication. By Theorem 3.6.12, it will suffice to prove the result when / E C~(IRN). For any (x,y,z) E ]R3, we have

If(x,y,z)1

=

liZ

0.=-00

aal (u,y,Z)dU/ S x

1

uER

IV/(u,y,z)ldu,

CHAPTER 3. MEASURES

112

and similarly,

1/(x,lI,z)1 $

r

l/(x,lI,z)1

IV/(X,l1,Z)ldl1, J veR Accordingly we can estimate 1/(x,,I,z)li

$

~

1

weR

IV/(X,II,W)ldw.,

(leR'V/(U,,I,z)ldU? x (leR IV/(X,l1,Z)ldl1) t x (LeR IV/(X,,I,W)ldW) t.

Integrating with respect to x and using Holder's inequality, we have

1

:teR

I/(x,y;Z)lidX$l :teRN

[(1

ueR

IV/(U''I,Z)ldU)! x

(leR IVf(X,V,Z)ldV) ' x (LER IV/(x,y,w)ldW) '] dx

=

(leit IV/(U,y,Z)ldU)! x

1 (( ·1~f(X'V'Z)ldV)' (1 x

:teR J veR

$

weR

IV!(x,y,W)ldW) 1 dx

(leR1V/(U,y,Z)ldu)' x

(1 /, .

(1 1

IV!(X,V,Z)ldl1dX) , x IV/(x,y,W)ldWdX) , . :teR veR zER wER Then we integrate with respect to y, again use Holder's inequality, and estimate

11 1 (1

,1eR :telR

lIeR

~

I/(x,y,z)I' dxdy

ueR

~

(1 1 (1 1 (11

IV/(u,y,z)1dU)' x

zeR lIeR

zER wER

(rJzeR JveR { IV/(X,V,Z)ldl1dx)t x

1

IV/(x,v,z)1 dVdX) , x

IV/(x,y,W)ldWdx) , dy

lIeR uER

IV/(u,y,Z)ldUdY) 1

{ { IV/(x,y,W)ldWdxdfl)i. J ,IeR Jz eR weR Finally, we integrate with respect to z, use Holder's inequality, and estimate X (

J. 1 1

zeR lIeR zeR

1/(x,y,z)I' dxdydz ~

3.7. DOMAINS WITH FINITE PERIMETER

r }zeR r 1weR IV/(Z,Y,W)ldWdxdY)! (}"eR

1 [( r 1 (1 1 aeR

}zeR tieR

x

IV/(Z,V,Z)ldVdx)! x

"eR ueR

~

113

IV/(U,y,Z)ldUdY)!] dz

(r}"eR }zeR r }weR r IV/(z,Y,W)ldWdxdY) 1

x

r 1tieR IV/(Z,V,Z)ldVdxdZ)! x (1zER}zeR

(1 1 1

aeR "ER uER

.

=

(L IV/ldC s

IV/(u,y,z)1 dUdydZ) 1

i

3)

•

•

Corollary 3.7.10 (Isoperimetric Inequality) 1/ S c JRH is a set 0/ finite perimeter and finite Lebesgue measure, then the following inequality is valid: (3.93)

Proof: Apply Lemma 3.7.8 to Xs.

•

Remark 3.7.11 Recall that Remark 3.87 showed us that the perimeter of a C l domain in RN agrees with the (N - 1)-dimensional Hausdorff measure of the topological boundary of the domain. Thus, when N = 2, the Isoperimetric Inequality gives a lower bound for the length required to enclose a given area, and when N = 3, the Isoperimetric Inequality gives a lower bound for the area required to enclose a given volume. Similar interpretations apply in higher dimensions. The version of the Isoperimetric Inequality given in Corollary 3.7.10 is not sharp: Simply consider the unit ball in llt2 for which the left-hand side of (3.93) equals 7r l / 2 and is strictly less than the right-hand side of (3.93) which equals 27r. A fairly simple geometric argument, which we sketch, allows us to obtain the following inequality as a corollary of the above lemma. We will call this a Poincare Inequality, because it applies to the unit cube (even though it is not for the L2 norm).

CHAPTER 3. MEASURES

114

Corollary 3.7.12 Let Q denote the closed unit cube in aN and suppose n is open with Q c n. There exists a constant C, depending only on. N., such that N-l

(k II - IQI7P:r deN)

l¥

~ C kIV/ld£.N:

(3.94)

holds lor Ie BV(n) , where

(3.95) Proof: Again it follows from Theorem 3.6.12 that it suffices to prove the result under the stronger assumption that I e C!(O). We define a function g: [-1,3]N -+ ]RN by reflecting through the faces ofthe unit cube. That is, g(XloX2,'" ,XN) = l(x~,x2'" "xN)' where X~ =

{

-Xi

if -1

~ Xi ~

Xi

if 0 ~

Xi

~

1

if 1 ~

Xi ~

2

Xi

-1

0

Since I is continuous, there is a point Xo e Q such that IQ = /(xo). Let Uzo : RN -+ [-~, 3]N be any smooth transformation that is the identity on Q and at distance more than 1/2 from Q maps all points to Xo. The result follows by applying Lemma 3.7.8 to the function (g 0 U) - I Q and using the change of variables formula for multiple integrals. This yields a constant C(xo) for which (3.94) will hold whenever IQ = I(xo). FUrther supposing that the transformation Uzo is defined in a way that depends continuously on xo, we see the value of C needed to make (3.94) generally valid is the maximum of the values C(xo), as Xo ranges over Q. • Remark 3.7.13 We have used the unit cube because the notion of reflection through a face is easily understood. Poincare's Inequality 3.7.12 is typically stated with Q a ball of radius r. An elementary change of variables shows that the specific choice of Q as a cube, ball, etc. only affects the value of the constant, not the basic validity of the result. One caveat is that if non-symmetrical figures (for example, ellipsoids) are used, then the ratio of the smallest dimension to the largest dimension will need to be bounded above and below. Using our version of Poincare's Inequality, we can prove a local form of the Isoperimetric Inequality wl.ich will be useful for analyzing the fine structure of sets of locally finite perimeter. Corollary 3.7.14 (Local Isoperimetric Inequality) 11 S c RN is a set with locally finite perimeter, then lor each X e aN and each r > 0 the

3.7. DOMAINS WITH FINITE PERlMETER

115

following inequality is valid:

in! { .cN(B(x,r)nS), .cN(B(x,r) \S) } :5 C [ 1I0SIIB(x,r)]

N/(N-l)

. (3.96)

Here C is a COMtant depending only on N.

Proof: Apply Corollary 3.7.12 to Xs, but with Q replaced by the open ball of radius r about x (as we may b~ Remark 3.7.13). • Structure Theory for Sets with Finite Perimeter In an earlier example, we showed that a domain with a C 1 boundary is a set of locally finite perimeter. The main goal of our st~y of sets of locally finite perimeter is to show the extent to which the Converse of this statement is true. In other words, we wish to show that a set of locally finite perimeter has a Cl boundary in a measure-theoretic sense. The term "boundary" here cannot mean the topological boundary; point set topology is too crude a tool to isolate the relevant points. Instead we must use the "reduced boundary" as defined next. DeJlnition 3.7.15 Suppose S C ]RN is a set of locally finite ,perimeter. The reduced boundary of S, denoted by 0" S, is the set of:r: e]RN such that (i) 1I0SII(B(x,r») > 0 holds for r > 0, •••

(11) "S(X) (iii)

.

=l,$

dll OS II 1I0SII(B(x,r» exists,

fB(:r,r) liS

IVsl = 1.

Theorem 3.7.16 If S is of locally finite perimeter, then the reduced boundary of S contains lIaSIl-almost all of S. Proof: This is a consequence of the Lebesgue differentiation theorem for • general Radon measures on ]RN. At a boundary point of a smooth domain, the boundary will infinitesimally be an N -1 dinlensional ball that separates a half-ball in the domain from a half-ball in the complement of the domain. Such behavior can be interpreted in terms of densities at the boundary point. In particular, each element of the boundary is also an element of the reduced boundary. In the next lemma, we obtain some density estimates that apply at each point of the reduced boundary. These estimates are step toward showing the reduced boundary is much like the boundary of a smooth domain.

a

CHAPTER 3. MEASURES

116

Lemma 3.7.17 There exists a constant 0 < C = C(N) < ex) such that for any set S with locally finite perimeter the following inequalities hold Jor x e 8·S:

(ii)

C- 1 ~liminfr.j.O r- N .cN(B(x,r)

\S),

(iii) C-I~liminfr.j.O r-(N-I) 118SII(B(x,r») ~ limsuPr.j.O r-(N-I)

118SII(B(x,r»)

~ C.

=

Proof: For convenience of notation, let us suppose that 0 x E ao S and that lISCO) eN. We apply Lemma 3.7.6 with 9 chosen to be the constant vector eN in some neighborhood of O. Then, for all sufficiently small r > 0, the left-hand side of (3.91) vanishes. Thus we conclude that for .c1-almost all sufficiently small r > 0

=

eN' (

iJ1(o,r)

"5 dll8S11

=

l. eN'''sdIl8SII.

JB(o,r)

=

-

(

~

llN - 1 [Sn§(0,r)].

JSnS(O,r)

Dividing both ends of the preceding formula by Definition 3.7.15(ii) implies

eN' ..I/.. aHN-1y 1111 '

118S11 (B(O, r») , we see that

1lN-l[Sn§(O,r)] 1 < II· m -__...:=:....,.-....:..".~ -

r.j.O

II 8S II (B(O,

r»

(3.97)

Notice that limsup r-(N-l) 118SII(B(0,r») ~ C r.j.O

follows from (3.97), proving the second half of part (iii). Note that Lemma 3.7.6 also implies that

p(snB(O,r)} $ II as II (B(O,r») +ll N - 1 [Sn§(0,r)]

(3.98)

holds for .c1-almost every r > O. Combining the Isoperimetric Inequality of Corollary 3.7.10 and equations (3.97) and (3.98), we see that for each 1 < ~ there exists 0 < r(~) such that

3.7. DOMAINS WITH FINITE PERlMETER

117

holds for .e 1-almost every 0 < r < r(~). Equation (3.99) is actually a slightly disguised differential inequality: Notice that if we set /(r) = .eN(SnB(O,r», then J'(r) = ?iN-l[S n S(O, r)].

Thus (3.99) tells us that (after we make a choice of constants)

C:5 [J(r)]

~

and rename the

*-1 J'(r) = N ! [J(r)] *

(3.100)

holds for .e 1 -almost all sufficiently small r > O. Part (i) of the lemma now follows by integrating (3.100). As an additional bonus we also obtain the inequality (3.101)

°

We obtain part (ii) of the lemma by noting that if is in the reduced boundary of S, then it is also in the reduced boundary of ]RN \ S. Finally, the first half of part (iii)·follows from parts (i) and (ii) and the Relative Isoperimetric Inequality. • Example 3.7.18 The theorem tells us that 1{N-l(8*S) :5 118SII(]RN), so for a set of finite perimeter we have (3.102) Equation (3.102) is false if 8* S is replaced by the topological boundary: For example if S is a union of balls centered at the points ·of a countable dense subset of lR(O, 1) and with radii chosen so that the total .eN measure of S is strictly less than that of iB(O, 1) and the sum of the 1{N-l measure of the surfaces of the balls is finite, then S will be of finite perimeter, but 8S will contain 1(0, 1) \ S, and thus have positive .eN measure and infinite 1£ N -1 measure. Notation 3.7.19 For a set sCaN and for r > 0, we set

rS

= {rx : XES}.

An important tool in proving the Structure Theorem for sets of finite perimeter is blowing-up3. Blowing-up produces local information that relates the vector to the geometry of S. The easy part of blowing-up is in the following lemma.

Vs

3This is not the blowing-up of algebraic geometry (as discussed in Krantz and Parks [2], for example).

CHAPTER 3. MEASURES

118

Lemma 3.7.20 118 is of locally finite perimeter, 0 e 8-8, and ri is a ~ with r. -+·00 as i -+ 00, then there e:r:Ut a subsequence ril and a set H of locally finite perimeter such that (3.103) and such that ""'1 s 118(ril 8)11 converges weakly to IIHII8HII.

Proof: The existence of H as in (3.103) follows from the Compactness Theorem. 3.6.r4 and the density estimate in part (iii) of Lemma 3.7.17. The weak convergence is a consequence of the theorem of de la Vallee Poussin that states that any sequence of uniformly bounded Radon measures has a subsequence that wealdy converges to Radon measure (a consequence of the Banach-Alaoglu Theorem, Rudin [3],3.15, or see Appendix A of Giusti [1] for a direct proof). • The geometric import of blowing-up is shown in the following theorem. Theorem 3.7.21 (Blowing-Up) If 8 is of locally finite perimeter, 0 E 8-8, and IIS(O) = eN, then X,.S

-+

XII-

in Lloc

as r

r .cN[nn+ n (111(0, R) n 8)] ATo RN = lim ~o

.cN(nn- n (B(O,R)

(3.104)

-+ 00,

0

(3.105)

,

\ 8)] = 0

RN

(3.106)

'

lim 118811111(0, R) = 1.

(3.107)

~o WN,....lRN 1

Here JH[- is the lower half-space {x : x· eN half-space {x: x·eN > O}

< O} and

JH[+ is the upper

Proof: We concentrate on (3.104). The other conclusions will then follow easily. It suffices to show that any sequence Ti -+ 00 has a subsequence Tis such that (3.108) Xr;s S -+ XHin Lloc as j -+ 00. By the lemma, we know there is a subsequence Tis such that (3.109) for some set H of locally finite perimeter and that additionally

converges weakly to IIHII8HII. We must show that H

= JH[-.

"'Ii ',?

3.7. DOMAINS WITH FINITE PERIMETER

119

First, we show that VH = eN for 118HII almost every point. For q, E C:(lRN ;IIlN ), we use the change of variables r'iY = x and the notation t/J(y) q,(r'JY) to see that

=

= jRN f divq,(x)Xr'Js(x)dCNx = r~-l jRN f divt/J(y)Xs(y)dCNy = r~-l

f t/J'

d118811.

VS

We conclude that IIOr,; SIIB(O, R) and

1

Vr, S

B(O,R)

;

=rr., 1 1l8811B(0, r:', N R)

118r.,SII = rr.- 1

l

'B(o,r:J- N

Vs R)

(3.110)

118811·

(3.111)

Because 0 E 8* S, (3.110) and (3.111) allow us to conclude that •

hm

dll8r., SII . iB(o,r:- N R) ,; = hm; 118r., 81115(0, R) j--+oo 118SI115(0,

iB(o R) V r, S

j--+oo

Vs

r:

dll8S11

i N

R)

= eN.

(3.112)

Considering R > 0 for which 118HII8B(O, R) = 0, we see by the Semicontinuity Theorem 3.6.5, by (3.112), and by the weak convergence of vr "s!l8(r.,S)1I to vHII8HII that 118HIli8(0, R)

:5

=

lir.:r IIOr., 811!(0.. R) .lim

J-+OO

~

ji(O,R)

eN' Vr,.s dll8r.,SII ,

= ji(O,R) ~ eN'vHd1l8HII· Since IVHI

= 1 holds 118HII-almost everywhere, we must have (3.113)

118HII-almost everywhere. While it seems intuitively clear that (3.113) implies that H is a halfspace with its bounding hyperplane orthogonal to eN, the point of this theorem is to prove that the vector really does have that desired geometric significance. Thus we argue as follows: Let t/Jtr be a family of molIifiers as in Section 1.3. For rP E (lRN ; lRN), we have

Vs

C:

-f

grad(t/Jtr*XH)"rPdc.N

=

f

t/Jtr*XHdivrPdCN

120

CHAPTER 3. MEASURES =

/ X.Hdiv(1/1" *4»dC N

=

/ eN· (1/1" * 4»

=

/ 1/1,,* (4). eN) d1l8HII·

dll8HII

We conclude that the smooth function 1/1" * XH is independent of the first N - 1 coordinates and is a non-increasing function of the Nth coordinate. Since 1/1" * XH converges CN -almost everywhere to XH we conclude that H is some translate of 1HI- in the eN direction. Finally, since Xr/.s converges weakly to XH in Lloc we see that parts (i) 1 and (ii) ofLemma 3.7.17 imply that indeed .J! = 0- . • The following lemma shows that if the results of blowing-up are sufficiently uniform then significant global information can be obtained. The proof of the lemma· is highly technical, but it seems to be an unavoidable fact of geometric analysis that a highly technical construction lies at the heart of any argliment going from local information to global information. Lemma 3.1.22 Suppose that

(i) K C JRN is compact, (ii) ,,: K -+ S(O, 1) is continuous,

Then there exist countably many Lipschitz functions Ii : IRN orthonormal bases {en, ei2, •.. , eiN }, i = 1,2, ... , such that

1

-+ IR and

00

Kc

UG

i

(3.114)

i=1

where (3.115)

Proof: First, we observe that, for each j, since K is compact, K is contained in finitely many open balls of radius 2- i centered in K. Taking the union of all these finite families and replacing each open ball by its closure, we construct a countable set B of closed balls centered in K such that, for each to > 0, (3.116) { B E B : diam(B):::: to } is finite

3.7. DOMAINS WITH FINITE PERIMETER

121

and K C U{ B

e 8 : diam(B) < E }.

(3.117)

Next, we construct a countable family V of orthonormal bases of IIlN such that the set consisting of the Nth vector from each basis in V forms a countable dense subset of 8(0,1). Thirdly, we arrange B x V x N into a sequence (3.118) { (B(Pi,Ti), {eil,ei2, ... ,eiN}, M;) }~ .

.=1

n i"(Pi, Ti) satisfying y). em] eiNI for all x, y e Gi. (3.119)

For each i, let Gi be a maximal subset of K I(x - y). eiNI :5 M;I{x - y) - [(x The function

Ii is defined by setting

Ii (x • eil,X' ei2,···,X 'eiCN-l)} = The domain of Ii is

X·

eiN for x

Ai = {(x·eu,x·ei2, ... ,x·eiCN-l)} : x

e Gi •

e Gi}.

{3.120}

{3.121}

We see that Ii is Lipschitz from the defining condition (3.119) for G i , and that the Lipschitz constant of Ii does not exceed Mi' We can extend the domain of Ii to all of IIlN- 1 by using the following formula (due to H. Whitney): (3:122) fi(x} = SUP{Ji(Z) - Mi dist(x, z} : z e Ai}. It remains to show that (3.114) holds. To this end, we argue by contradiction and assume to the contrary that (3.114) does not hold. Suppose x· e K, but x· 'I. G i , for i = 1,2, .... By (3.116) and (3.117), there must be infinitely many balls in B that contain x·. Set 8' = {B e 8 : x· e B}. We now choose a sequence in 8' x V x N with some particular properties: Let i j be such that that x· e B(Pii' Ti;} and

= 0,

(3.123)

.lim ei;N = v(x·},

(3.124)

.lim

{3.125}

lim Ti'J

j .... oo

J .... OO J .... OO

MiJ

=

00.

'I. Gi;, there must be Yi e G iJ C K n i"(Pi;' Ti;}

Now, for each j, since x·

(3.126)

such that I(x· - Yi}' ei;NI > Mi; I{x· - Yi} - [(x· - Yi)' ei;N] eiJNI. Equations (3.123}-(3.127) contradict part (iii) of the hypotheses.

(3.127)

•

CHAPTER 3. MEASURES

122

Remark 3.7.23 The functions Ii defined in the proof of Lemma 3.7.22 can be extended to be Cl functions, instead of just Lipschitz functions, if we are willing to appeal to the Whitney Extension Theorem. While the Whitney Extension Theorem is not proved until Chapter 5, its proof is independent of the results of this section. Measure theory allows the local information provided by blowing-up to be made sufficiently uniform that the preceding lemma can be applied. The result is the following theorem.

Theorem 3.7.24 (Structure Theorem) If 5 c aN hal locally finite perimeter, then there cist countably many (N -1) -rectifiable seu, G1 , G2 , ••• and a set E fllith 118SII(E) = 0 such that 00

8*5 = Eu

UG•.

(3.128)

i=1

Proof: Egoroff's Theorem (see Evans and Gariepy [1], Section 1.2, or Federer [4], 2.3.7) tells us that, for each f > 0, pointwise convergence of measurable functions implies uniform convergence off a set of measure f. Applying Egoroff's Theorem countably many times, we see that there exist disjoint 118SII-measurable subsets of 8* S, covering 118SII-almost all of 8* S, on which the convergence in (3.105)", (3.106), and (3.107) is uniform. Similarly, Lusin's Theorem (see Evans and Gariepy [1], Section 1.2, or Federer [4], 2.3.5) tells us that, for each f > 0, a measurable function is equal to a continuous function off' a set of measure f. Applying Lusin's Theorem countably many times on each of the preceding subsets, we see that each is 118SII-almost the union of disjoint compact subsets on which the normal vector liS is a continuous function. We wish to apply Lemma 3.7.22 to each of these compact subsets. To do so, we need to verify that hypothesis (iii) of Lemma 3.7.22 is satisfied, but the density estimates of Lemma 3.7.17 insures that (3.105) and (3.106) imply that hypothesis. • Remark 3.7.25 As in Remark 3.7.23, the Whitney Extension Theorem allows one to conclude that the sets G. can be required to be subsets of the images of C 1 functions. The final result of this section gives a very general form of the GaussGreen Theorem. This result was discovered independently by E. De Giorgi and H. Federer.

Theorem 3.7.26·(Generalized Gauss-Green Theorem) If has finite perimeter, then for each l/J E C: (JRN ; aN) we have

1

divl/Jdr.N = [

S

18 • s

l/J. liS rIH. N -

1•

sCaN (3.129)

3.8. THE AREA FORMULA

123

Proof: We need to show that on 0° S the measure II oS II equals the N - 1 dimensional Hausdorff measure. This follows from (3.107) and the Structure Theorem 3.7.24.

•

Remark 3.7.27 lithe set S in Theorem 3.7.26 is not of finite perimeteror even locally finite perimeter-one can still use Equation 3.129 as the definition of the boundary. Of course, it is more interesting if both sides of the equation can be given by independent definitions .. We direct the reader's attention to just such a non-trivial extension of the Gauss-Green Theorem that has been given in Harrison and Norton [I].

3.8

The Area Formula

The main result of this section is the following theorem. Theorem 3.S.1 (The Area Formula) If f : function and M ~ N, then

]RM

-t ]RN is a Lipschitz

r JMf(x)dCMx= iRNr card(Anrl(y»dllMy

iA

(3.130)

holds for each Lebesgue measurable subset A of]RM.

Here 1£ M denotes the M -dimensional Hausdorff measure and JM f denotes the M-dimensional Jacobians of f which is defined in Definition 3.8.3. In case M = N, the M-dimensional Jacobian agrees with the usual Jacobian, Idet(Df)I· The proof of the area formula separates intt> three fundamental parts. The first is to understand the situation for linear maps. The second is to extend our understanding to the behavior of maps that are well approximated by linear maps. This second part of the proof is essentially multi-variable calculus, and the area formula for Cl maps follows readily. The third part of the proof brings in the measure theory that allows us to reduce the behavior of Lipschitz maps to that of maps that are well approximated by linear maps. In the next section we will treat the co-area formula that applies to a Lipschitz map f : ]RM -t ]RN, but with M 2': N instead of M ~ N. The proof of the co-area formula is similar to the proof of the area formula in that the same three steps of understanding linear maps, understanding maps well approximated by linear maps, and applying measure theory are fundamental. The discussion of linear maps in this section will be applicable to both the area formula and the co-area formula. Linear Maps

A key ingredient in the area formula is the K -dimensional Jacobian which

CHAPTER 3. MEASURES

124

is a measure of how K -dimensional area transforms under the differential of a mapping. Since a linear map sends one parallelepiped into another, the fundamental question is "What is the K -dimensional area of the parallelepiped determined by a set of K vectors in ~ 1" Of course the answer is known, but recently G. J. Porter [1] has discovered a particularly lucid derivation that we give below.

Proposition 3.8.2 If

Vii V2i

tt i = (

:

)

'

for i

= 1,2, ... ,K,

(3.131)

VNi

are vectors in ]RN, then the pamllelepiped determined by those vectors has K -dimensional area (3.132)

where V is the N x K matrix with

tt 10 112 , ••• ,11K

as its columns.

Proof: H the vectors ttl, tt 2, ... , tt A: are orthogonal, then the result is immediate. Thus we will reduce the general case to this special case. Notice t~at Cavalieri's Principle shows us that adding a multiple of tti to another vector tt j, j =I i, does not not change the K -dimensional area of the parallelepiped determined by the vectors. But also notice that such an operation on the vectors 11i is equivalent to multiplying V on the right by a K x K triangular matrix with l's on the diagonal (upper triangular if i < j and lower triangular if i > j). The Gram-Schmidt procedure is effected by a sequence of operations of precisely this type. Thus we see that there is an upper triangular matrix A with 1's on the diagonal such that V A has orthogonal columns and the columns of V A determine a parallelepiped with the same K -dimensional area as the parallelepiped determined by 111, 11 2 , ... , 11 N. Since the columns of V A are orthogonal, we know that Jdet (V A)t • (V A» equals the K-dimensional area of the parallelepiped determined by its columns, and thus equals the K-dimensional area as the parallelepiped determined by tt 10 11 2 , ••• ,11N. Finally, we compute

det(VA)t.(VA))

=

det(At.Vt.V.A)

= =

det

(At) det (V t •v) det(A) det (v t . V) .

•

3.B. THE AREA FORMULA

125

Definition 3.8.3

(i) SUppo$e U C RM and f : U ~ RN. We say J is differentiable at a e U iJ there exists a linear map L : RM ~ RN such that Ii IJ(x) - J(a) - L(x - a)1 - 0 :c~ Ix-al - .

(3.133)

In CtJ8e / is differentiable at a, we denote the linear map L such that (3.133) holds by DJ(a) and call it the differential of J at a.

(ii) Suppose that J : RM ~ RN is differentiable at a and K :5 M. We define the K-dimensional Jacobian of J at a, denoted JKJ(a), by setting

{1lK[D J(a)(P)] . J /( ) _ K a - sup 1l K [P] . P is a K-dimensional parallelepiped contained in RM}. (3.134)

The conventional situation considered in elementary multi-variable calculus is that in which K = M = N. In that case, it is easily seen from Proposition 3.8.2 that one may choose P to be the unit M -dimensional cube and that hJ(a) = Idet(D/(a»I. Two other special cases are of interest: They are when K = M < N and when M > N = K. When K = M < N, again one can choose P to be the unit M-dimensional cube in RM. The image of P under DJ(a) is the parallelepiped determined by the columns of the matrix representing DJ(a). It O

follows from Proposition 3.8.2 that h J(a) = Jdet [(DJ(a»t o· (D/(a»]. When M > N = K, then P should be chosen to lie in the orthogonal complement ofthe kernel of D J (a). This follows from Corollary 3.1.17 which tells us that orthogonal projection onto an affine subspace of IRN cannot increase Hausdorff measure. So, if we let L be the orthogonal complement of the kernel of DJ(a), then if pI is any parallelepiped, the image of the orthogonal projection of pI onto L is the same as the image of pI; but the orthogonal projection of pI has smaller K -dimensional area, giving a larger ratio of areas. It is plain to see that the orthogonal complement of the kernel of D J(a) is the span ofthe columns of (D J(a»t. If we begin with the parallelepiped determined by the columns of (DJ(a»t, then that parallelepiped maps onto the parallelepiped determined by the columns of (D /(a»· (D J(a»t. By Proposition 3.8.2, the N-dimensional area of the first parallelepiped is

J

det [(D J(a» • (D J(a»t]

CHAPTER 3. MEASURES

126

and the N-dimensional area of the second parallelepiped is det [{(DI(a». (D I(a»t) t • (DI(a». (DI(a»t)]

= det [(D I(a» • (D I(a»t] , so the ratio is JKI(a) = Jdet [(DI(a». (DI(a»t]. We summarize the above facts in the following lemma. Lemma 3.8.4 Suppose

(i)

I : lllM -+ lllN

II K = M = N, then

= Idet(DI(a))l.

(3.135)

= Jdet [(D!(a»t.(Df(a»].

(3.136)

JKf(a) = .jdet [(Df(a»·(Df(a»t].

(3.137)

JKI(a)

(ii) If K = M

~

N, then JKf(a)

(iii)

is differentiable at a.

II M

~

N = K, then

Remark 3.8.5 The Generalized Pythagorean Theorem from Porter [1] allows one to see that the right-hand side of either (3.136) or (3.137) is equal to the square root of the sum of the squares of the K x K minors of D f( a), where K = min{M,N}. This is the form one is naturally led to if one develops the K-dimensional Jacobian via the alternating algebra over lllM and RN as in Federer [4]. We will also need to make use of the polar decomposition of linear maps. Theorem 3.8.6 (Polar Decomposition)

(i) II M ~ Nand T : lllM -+ lllN is linear, then there exists a symmetric linear map S : lllM -+ JRM and an orthogonal linear map U : RM -+ lllN such that T = U 0 S. (ii) II M ~ Nand T : lllM -+ lllN is linear, then there exists a symmetric linear map S : JRN -+ RN and an orthogonal linear map U : JRN -+ RM such that T = S 0 U t . Proof: (i) For convenience, let us first suppose that T is of full rank. The M x M matrix Tt • T is easily seen to be symmetric and positive definite. So

3.8. THE AREA FORMULA

127

Tt •T has a complete set of M orthonormal eigenvectors ttl, tt 2, ••• ,11M associated with the positive eigenvalues ~1' ~2' ••• , ~M.

We define 8 : aM -+

aM

by settulg .

Using the orthononnal basis 111 , 112, ... ,11M , we see that 8 is represented by a diagonal matrix, thus S is symmetric. We define U : JRM -+ aN by setting 1

u(lI i ) = .;Ai T(lI i ). We calculate

Thus U is an orthogonal map. In case T is not of full rank, it follows that some of the ~i 's may be zero. For such i's we may choose U(tt.) arbitrarily, subject only to the requirement that U(lI 1), u(lI 2)"'" u(lI n) b~ an orthonormal set. (ii) We apply (i) to the mapping Tt to.2btain a symmetric 8 and orthogonal U so that Tt = U 0 8, but then T = (U 0 8)t = 80 Ut .

•

Remark 3.8.1 The equations in Lemma 3.8.4 can be obtained using the Polar Decomposition of linear maps. For example, if we are considering f : liM -4 liN with M :5 N, then we apply the Polar Decomposition to write D f(a) = U 0 S. Since an orthogonal map clearly preserves Hausdorff measure, the entire effect of D f(a) on the Hausdorff measure of a parallelepiped in liM must be due to the non-negative definite symmetric map 8, so we have JMf(a) = det(S). But then we can compute det(8)

=

Vdet(St)det(S)

=

Vdet(St. ut·U ·8) = Vdet [(Df(a»t • (Df(a))).

CHAPTER 3. MEASURES

128

The first application of the Jacobian is in the following basic lemma the behavior of Lebesgue measure under a linear map.

concernin~

Le~ 3.8.8 If A c aM is Lebesgue measurable and T : aM ~ aM is linear, then .eM (T(A» = Idet(T)I .eM (A).

Proof: Given ~ > 0, we can find an open U with A c U and .eM (U\A) < ~. We subdivide U into cubes and the image of each cube is a parallelepiped. So .eM (T(A)) :5 .eM (T(U» :5 Idet(T)I.eM (U) :5 Idet(T) I [~+ .eM (A)].

Letting

~

..l. 0, we see that .eM (T(A» :5 Idet(T)1 .eM(A).

Now we need to prove the reverse inequality. Note that if det(T) = 0, then we are done. Assuming det(T) :I 0, we apply the case already proved to T(A) and T- 1 to see that .eM (A) = .eM (T- 1(T(A))) :5 Idet(T- 1 )1 eM (T(A».

The result follows since det(T-1)

= (det(T»-l.

•

The Main Estinlates for the Area Formula Lemma 3.8.9 (Main Estinlates for the Area Formula) Suppose T : IRM ~ IRN is of full rank and 0 < ~ < ~. Let n be orthogonal projection onto the image of T. Set

A = inf{(T,v):

Ivl = I}.

(3.138)

If the Lebesgue measurable set A C IRM is such that

(i) Df(a) exists for a E A, (ii) IIDf(a) - Til <

~

holds for a E A,

(iii) If(y) - f(a) - (D f(a), y -

a)1 < ~ Iy - al

holds for y, a E A,

(iv) IIlf(A) is one-to-one, then (1 - 3~A _l)M . JMT· eM (A) :5 l£M (J(A»

:5 (1

+ 2~A-1)M ·JMT . .eM(A).

(3.139)

3.8. THE AREA FORMULA

129

Proof: First we bound l£M (t(A» from above. We use the polar decomposition to write T = U 0 S, where S : RM -+ RM is symmetric and U : RM -+ aN is orthogonal, and we note that S is non-singular with JMS = JMT and with ~-1 = liS-III. Set B = SeA) and 9 = loS-I. We know that £M(B) = JMSo£M(A)

We claim that

= JMLo£M(A).

Lip(gIB) $ 1 + 2£~ -1.

To see this, suppose z, be B. Then with a = S-l(b), y that Iy - al $ ~-llz - bl. Therefore we have

= S-l(Z), it follows

Ig(z) - g(b)1 $ Ig(z) - g(b) - (Dg(b), z - b)1 + I(Dg(b) - U, z - b)1 = If(y) - I(a) - (DI(a),y - a)1 +1«DI(a) - T) 0 5- 1 , Z - b)1 + Iz - bl $ t: Iy - al + liD I(a) - TlloliS-lUolz - bl + Iz - bl $ (1 + 2t:~ -1) Iz - bl.

+ I(U, z -

b)1

(3.140)

Finally, we haye

1lM (t(A»

=

'liM (g(B»

$

(1 + ~t:~-I)M o£M(B)

=

(1 +.2t:~-l)M oJMTo£M(A).

Next we bound l£M (t(A» from ·below. We continue to use the same notation for the polar decomposition. SeCV = IT(J(A» = IT(g(B» and h = (IT 0 9IB)-l. We claim that

Lip(hlc) $ (1- 3t:~-l)-l. To see this, suppose w, c e C. Let b e B be such that IT 0 g(b) = c and z E B be such that IT 0 g(z) = w. Arguing as we did to obtain the upper bound (3.140), but with some obvious changes, we see that Ig(z) - g(b)1 ;::: (1 - 2£~-l) Iz - bl.

Also, we have t:A-llz - bl

~

=

Ig(z) - g(b) - (Dg(b),z - b)l IIT(g(z) - g(b) - (Dg(b),z - b»

~

IIT.L(g(z) - g(b) - (Dg(b),z - b»)I

=

IIT.L(g(z) - g(b»I.

+IT.L (g(z) - g(b) - (Dg(b), z - b»

I

CHAPTER 3. MEASURES

130 Thus we have ID(g(z» - D(g(b»!

~ ~

Ig(z) - g(b)I-ID.L(g(z) - g(b» I (1- 2E~-1) Iz - bl- E~-llz - bl.

Finally, we have J MT • 1/. M (A)

=

CM(B)

::; (1- 3E~-1)M .£M(C) ::; (1- 3E~-I)M ''H.M(J(A»).

•

Rademacher's Theorem Theorem 3.S.10 (Rademacher's Theorem) A Lipschitz function 1 -+ JRN is differentiable C M -almost everywhere and the differential is a measumble function. "

aM

Proof: We may assume N = 1. We use inductiop on M. In case M = 1, the result follows from the classical theorem that an absolutely continuous function from IR to IR is differentiable C-almost everywhere. For the induction step M > 1, we see by Fubini's Theorem and the theorem about absolutely continuous functions that all M partial derivatives of 1 exist CM-almost everywhere and are "measurable functions. The goal is to show that these partial derivatives actually represent the differential at almost every" point. Let us write IRM = IRM - 1 X IR and denote points x E IRM by x = (y, z), 1/ E IRM-l, Z E R We consider a point xo = (1/0, %0) where the following two conditions are satisfied:

(i) As a function of the first M - 1 variables, f is differentiable.

(ii) All M partial derivatives of 1 exist and are approximately continuous (see Definition 3.5.7). For convenience of notation, we assume xo = (0,0) and that all the partial derivatives at xo vanish. It suffices to show that, for each E > 0, there exists r > 0 such that

I/(y,z)1 ::; ET holds whenever I(y, z)1 < T. Let E > 0 be arbitrary. We can choose TO > 0 so that, for Iyl < TO, I/(y,O)1 ::; Elyl. Choosing TO smaller if necessary, we can also assume that, for 0 < T < TO, the set of points in the cube [-T, r]M for which the absolute value of the Mth partial derivative exceeds E is less than EM TM •

3.B. THE AREA FORMULA

131

Inside any (M - I)-dimensional cube rr:!~l[Yi - fr,yd of side fr there must be a vertical line, say at y', in which the Mth partial derivative is bounded by f except for a set of length £r. "So I/(y',z)1

~ I/(Y',~)I + L% I:~

(y',()1

dC(

ely'l + elzl + Mer

~

and thus I/(y,z)1

~

I/(y',z)I+Mv'M-1fT

~

(2 + (1 +';M -l)M)fr.

•

Measure Theory of Lipschitz Maps

We need to know that the images of Lipschitz maps are measurable. Recall from the discussion of Borel sets versus Suslin sets in Section 1.3 that the properties of a set can be significantly altered by applying a function to it. Lemma 3.8.11 Suppose K < M and suppose I: JRM -+ JRN is a Lipschitz function. II A C IRM is llK -measurable and illl~ (A) < 00, then I(A) is llK -measurable.

Proof: Since A can be written as the union of countably many bounded measurable sets, it will suffice to prove the lemma under the additional assumption that A is bounded. Note that sets of llK measure zero are preserved under I. Since Hausdorff measure is Borel regular (by Lemma 3.1.2), it follows that A is contained in a Borel set B with llK (A) = llK (B). Note that (f(B) \ I(A» c I(B \ A) and llK (f(B \ A» = o. Thus, it will suffice to show that I(B) is ll K-measurable whenever B is a bounded Borel set with llK(B) < 00. Also because llK is Borel regular, a bounded Borel set B with llK (B) < 00 is llK-almost equal to a countable union of compact subsets of B. So the I-image of B is llK-almost equal to a countable union of I-images of compact sets. Of course, the I-image of a compact set is compact, hence ll K-measurable, and the result follows. • Lemma 3.8.12 Suppose K < M and suppose I : JRM -+ JRN is a Lipschitz function. II A c IRM is llK -measurable and illlK (A) < 00, then the function on IRN defined by

(3.141)

is llK measurable, and {

JRN holds.

card(A n 1-1 (y» d1l K

~ [Lip 11K llK (A)

(3.142)

CHAPTER 3. MEASURES

132 Proof: For each i > 0, write

where the sets Ai,j, j = 1,2, ... , are pairwise disjoint l£K-measurable sets all having diameter less than l/i. Then

is l£K-measurable by Lemma 3.8.11 and LX/(A;.;) (y):5 card(Anrl(y» for each y E JRN. j

Here LX/(A;,;)(Y)

E JRN, as i ~ 00,

t card(Anrl(y» for each Y

j

proving the 1l K-measurability of the function in (3.141). Finally, (3.142) follows from the Monotone Convergence Theorem and the estimate .

L RN

L

X I(A;,;) d1{.K

j

=

L

l£K[/(A i ,j)]

j

:5 [Lip I]K L l£K (Ai,j) j

= Corollary 3.8.13 For

[Lip I]K l£K (A).

•

I and A as in the preceding lemma,

1lK({y: rl(y)nA is infinite})

= O.

Proof: Equation (3.143) follows immediately from (3.142),

(3.143)

•

In the next few lemmas, we will show how to use the lexicographic ordering on JRM to divide a set into pieces on which the restriction of I is one-to-one. Definition 3.8.14 The lexicographic ordering, ~i, on JRM is given by defining (X},X2"",XM) -
if and only if, for some j E {I, 2, ... , M), Xl = Yl, X2 = Y2, ... , Xj-l = < yj. For X, y E aM, we will write X ~t Y if X -
Yj-l, and Xj

3.8. THE AREA FORMULA

133

Lemma 3.S.15 Suppose I : RM .-. R.N is a Lipschitz function. is eM -measurable, then, lor each i = t, 2, ... ,

Aj

II A c RM

i }

=

{x e A : card(rIV(x)] n A) =

=

I-I ({y e RN : card(f-I(y) n A)

= i })

is eM -measurable. Proof: It

will suffice to show that, for each positive integer i Ai = {x e A : card(rIV(x)] n A) ~ i

~

2,

}

is eM -measurable. It is also sufficient to prove this assertion under the additional assumption that A is bo\lDded. Fix i ~ 2. Define the Borel set S by setting

_ ==

(RM)2jn{(Ut.U2, ... ,Uj,X1.X2 ... ,Xj): Ui1 =1= Ui.. .XI

Define A C (RM)

for 1 ~ il < i2 ~

i,

= X2 = ... = Xj}.

2j by setting

A = {(Ul, U2,· .. , Uj'/(Ul), l(u2)"'" f(uj» : Ul> U2, ... , Uj e A}. Since A is the image of A under a Lipschitz mapping, it follows that A has . finite 1-lM measure and is 1-lM -measurable by Lemma 3.8.11. Now notice that Ai = II(S n A) where II : (RM) 2j -+ RM is projection onto the first factor. Since II is also a Lipschitz map, we see that Lemma 3.8.11 implies that Ai is eM-measurable. • Remark 3.S.16 Note that the preceding lemma gives us an alternative proof that if f : RM -+ RN is a Lipschitz function and A C RJI·[ is M _ measurable, then the function on RN defined by y 1-+ card(Anf-l(y» is 1-f.M -measurable.

e

Lemma 3.S.17 Suppose I : RM -+ RN is a Lipschitz function. If A C RM is eM -measurable, .then there exist £M -measurable sets Aj,k C A, i = 1,2, ... , k = 1,2, ... ,i such that j

Aj

=

UAj,k'

"=1 Here Aj is as in Lemma 3.8.15; also, for each k one-to-one, and I(A j ,")

e

{I, 2, ... ,i},

= {y e RN : card(rl(y) n A) = i }.

IIAj ...

is

CHAPTER 3. MEASURES

134

Proof: It is no loss of generality to assume that A = f- 1

({y e RN : card(r1(y) n A) = j

}).

We can then argue by induction on j. The case j = 1 is, of course, trivial. Suppose j ~ 2. Define the Borel sets 3., i == 1,2, ... ,M, by setting

-

(M)4n {(U,V,X,Y):U1=Vlt ... ,Ui_1 =Vi-1,U.
'::'i= R

x=y},

for i = 1,2, .. . ,M. Let n: (RMt _ RM be projection on the first factor. Define A C (RM) 4 by setting

A = {(u,v,f(u),J(v» : u,v e A}. Since A is the image of A under a Lipschitz map, A is ?tM-measurable by Lemma 3.8.11. Set !vi

B=

UII(:::. nA). i=l

Each set II(ZinA) is .eM-measurable because it is the orthogonal projection of a measurable set. Observe that; B consists of all points u e A for which there exists a point v e A that is larger than u in the lexicographic ordering on RM. Setting A;,l = A \B, we obtain an .eM -measurable set containing the lexicographically largest point of An f-1(y) for each y e f(A). Notice alSo that, on B, f is a (j - 1)-to-one function, so the induction hypothesis applies to B. • Lemma 3.8.18 Suppose f : RM - RN is a Lipschitz function and A C RM. If y e RN is such that f-1(y) n A is infinite, then every point of

f-1(y)

n {x:

Df(x) exists and is non-singular} nA

is an isolated point of that set.

•

Proof: Obvious. Lemma 3.8.19 Suppose f: RM is .eM -measurable and

B

=

-+

RN is a Lipschitz function. If A C RM

r1({y:r1(y)nA isinfinite}) \( {x: JMf(x) = O} U {x: Df(x) does not exist}),

has positive .eM measure, then there exists B' that fiB' is one-to-one.

c

B with .eM (B') > 0 such

3.8. THE AREA FORMULA

135

Proof: Choose a compact C C B with positive measure. Then, by Lemma 3.8.18, 1-1 [/(x)] is finite for each x e C. We can apply Lemma 3.8.17 to decompose C into countably many sets Ci ,,., at least one of which must have positive measure. • Proof of the Area Formula We first prove the result under the additional assumptions that DI(a) exists at every point a e A, that JMI(a) > 0 at every point of A, and that IIA is one-to-one. -By Lusin's Theorem'1.3.12 we may assume that DI(a) is the restriction to A of a continuous function. By Egoroff's Theorem 1.3.13 we may suppose that

I/(y) - I(a) - (DI(a),y - a)1

Iy-al converges uniformly to 0 as yeA approaches a e A. It is plain that, for any E > 0, every subset or" A with sufficiently small diameter satisfies conditions (i)-(iii) of Lemma 3.8.9 for some full rank linear T : JR.At _ JR.N -namely, - we can choose T to be D I at any point in such a sufficiently small set. Since we are assuming that D I on A is the restriction of a continuous function, we can find a positive lower bound for>. in (3.138). To see that condition (iv) of Lemma 3.8.9 is also satisfied on a subset of A of small enough diameter, we suppose II 0 I(y) = II 0 I(z); we show that, in this case, E > 0 can be chosen small enough compared to >. that conditions (i)-(iii) lead to a contradiction. Using (i)-(iii), we estimate I(T, y -

z)1 ~

~

III {T,y - z»1 III (T - Df(a),y - z)1 + III (Df(a) - DI(z),y - z)1 +III (DI(z),y - z)1 liT - DI(a)lIly - zl + IIDf(a) - DI(z) II Iy - zl +111 (Df(z),y - z)1

=

liT -

DI(a)lIly - zl + IIDf(a) - DI(z)lIly - zl +III(f(y) - fez) - (DI(z),y - z»1

~

liT -

DI(a)lIly - zl + IIDf(a) - DI(z)lIly - zl +I/(Y) - fez) - (DI(a),y - z)l·

By choosing a, y, z in a small enough set we can bound the right-hand side of the preceding inequality above by 3E Iy - zl, while the left-hand side is bounded below by >'Iy - zl. Choosing Esmaller than ~ gives a contradiction. Thus (iv) also must hold on subsets of small enough diameter. In case I is finite-to-one almost everywhere, we use Lemma 3.8.17 to decompose A into the sets Ai," on which the previous case applies.

1

136

CHAPTER 3. MEASURES

For the part of A on which I is infinite-to-one we know the right-hand side of (3.130) is zero by Corollary 3.8.13. H the left,:,~d side of (3.130) were not zero, we would contradict Lemma 3.8.19. Finally, to complete the proof, we need to shdW that the image of a set on which J M I = 0 has measure zero. That follows by defining I~ : RM -+ RM+N by "

Xt-+ (Ex,/(x»). This gives us the full rank hypothesis, but only increases the Jacobian by a bounded multiple of E. The image of I is the orthogonal projection of the image of IE and thus its Hausdorff measure is no larger than the Hausdorff measure of the image of I~. We conclude as E 1 0 that the Hausdorff measure of the image of I is O. •

3.9

The Co-Area Formula

The main result of this section is the following theorem. Theorem 3.9.1 (The Co-Area Formula) II I : RAt chitz lunction and M ~ N, then

-+

RN "is a Lips-

(3.144) holds lor each Lebesgue measurable subset A o/]RAt. Here JNI denotes the N-dimensional Jacobian of I which was defined in the previous section in Definition 3.8.3, and which was seen by "(3.137) to be given by

JNI(a) = "Jdet [(DI(a»· (DI(a»t]. In case M = N, the N-dimensional Jacobian agrees with the usual Jacobian, Idet(Df) I, and the area and co-area formulas coincide. In case M > N, and I : RAt = RN X RM-N -+ RN is orthogonal projection onto the first factor, then the co-area formula simplifies to Fubini's theorem, thus one can think of the co-area formula as a generalization of Fubini's theorem to functions more complicated than orthogonal projection. The co-area formula was first proved in Federer [3]. As in the proof of the area formula, the proof of the co-area formula separates into three fundamental parts. The first is to understand the situation for linear maps. This was done in the previous section. The second part is to extend our understanding to the behavior of maps that are well approximated by linear maps. The third part of the proof brings in the measure theory that allows us to reduce the behavior of Lipschitz maps to that of maps that are well approximated by linear maps.

3.9. THE CO-AREA FORMULA

137

Main Estimates for the Co-Area Formula Lemma 3.9.2 (Main Estimates for the Co-Area Formula) Suppose M > N, U : RN -+ RM is orthogonal, and 0 < E < Lebesgue measurable set A C RM is such that

l.

If the

(i) Df(a) exists for a E A,

(ii) IIDf(a) - Utll <

E

holds for a E A,

(iii) If(Y) - f(a) - (D f(a), y - a}1 <

ElY - al

holds for y, a E A,

then (1- 2E)M {

JaN

1tl"I-N(Anrl(y))dCN y

:5 (

JRN

~

1 A

JMf(a)dCMa

'HM-N(Anf-l(y» dCNy.

.

(3.145)

Proof: Let V : RM-N -+ RM be an orthogonal map such th~t ker(Ut ) and ker(Vt) are orthogonal complements. Defule F : RM -+ RN X RM-N by setting F(x) = (i(x), vt(x)) , and let II : ]RN x RM-N to see that

-+

]RN be projection on the first factor. It is easy

Subsequently, we will show that FIA is one-to-one so that, by the area formula, CM[F(A)] =

L

JMFdC M

=

L

JNfdC M .

Thus, using Fubini's Theorem, we h!l.ve CM[F(A)] (

JRN

'HM-N[F(A)

n II-1(z)] dCN z

( 'HM-N[F(Anf-l(z»]dCNz. JRN To complete the proof, we show FIA to be one-to-one and estimate the Lipschitz constant of F on Anf- 1 (z) and the Lipschitz constant of F-l on F(A n f- 1 (z». Suppose a, yEA n r 1 (z). Then F(a) = (/(a), Vt(a» =

CHAPTER 3. MEASURES

138

(z, vt(a» and F(y) = (f(y), vt(y» = (z, vt(y». We should like to comla - yl and IF(a)·- F(y)l. But the first components are the same,

pare so

IF(a) - F(y)1

= IVt(a) -

vt(y)l.

On the one hand, V t is distance decreasing, so

IF(a) - F(y) I ~

la - yl·

On the other hand,

IWt,y - a}1

~

I(DI(a),y- a}l + IIDI(a) -

<

2Ehi -ai,

I/(Y) - I(a) - (DI(a),y -

Utll Iy - al a}l + IIDI(a) - utll Iy - al

and

so Thus we have

VI -

4E2

Iy - al ~ IF(y) -

F(a)1 ~

Iy - al·

•

Corollary 3.9.3 $uppose M > N, T : ]RM -+ ]RN is 01 rank N, and 0< E < II the Lebesgue measurable set A C ]RM is such that

!.

(i) DI(a) exists lor a (ii) IIDI(a) - Til <

E

E

A,

holds lor a E A,

(iii) I/(Y) - I(a) - (DI(a), Y - a)l < ElY - al holds lor y, a E A, then (1- 2E)M

f

JRN

llM-N(Anrl(y» dCNy

~!.

aN

~ f htf(a)dCMa

JA

1i M - N (A n 1-1 (y») dCN y.

(3.146)

Proof: By the Polar Decomposition Theorem 3.8.6, there exists a symmetric linear map S : ]RN -+ lItN and an orthogonal map U : ]RN -+ ]RM such

3.9. THE CO-AREA FORMULA

that T = So ut. Set 9 = S-1 obtain

(1- 2f)M

r

JRN

0

139

f. Then we apply the lemma to 9 and U to

1t M - N (Ang- l (z»d,CNz

~

~ f

JA

JMg(a)d,CMa

r 1tM-N(Ang-l(z» dCN z.

JRN

(3.147)

Notice that if y = S(z), then

so by the change of variables formula in R,N applied to the mapping S, we have

Also we have JNS JMg = JMf; so

L

JNgJMg(a)d,CMa=

holds. Thus if we multiply all three (3.146).

L

JMf(a)dCMa

te~.in

(3.147) by JNS, we obtain •

Measure Theory of Lipschitz Maps We need to verify that the integrand on the right-hand side of (3.144) is measurable. (The measurability of the integrand on the left-hand side of (3.144) is given by Rademacher's Theorem 3.8.10.) First we obtain a useful preliminary estimate originally proved by Ellenberg and Harrold. Lemma 3.9.4 Suppose 0 ~ N :$ M < 00. There exists a constant C(M, N) such that the following statement is true: II I : liM -+ liN is a Lipschitz /unction and A C liM is'c M -measurable, then

holds. Proof: We may assume the right-hand side of (3.148) is finite. Fix u > O. By the definition of Hausdorff measure, there exists a cover of A by closed sets S1. S2, ... , all having diameter less than u, such that

~ TM •

(diar;(Si»)

M

:$

llM (A)

+ u.

140 For y

CHAPTER 3. MEASURES E .]RN

we observe that

Note also that if pES;, then !(S;)

c i(!(P), Lip(f) diam(S;») ,

so

Thus we have

r

iRN ~

1i~-N (A n rl(y» dCN y 2N - M "fM_N E(diam(S;»)M-N {

iRN

i

~

2N - M "fM_N"f N [Lip(f)]N

~

2N

. .

X/(s;)dC N

E (diam(S;»)N i

™:r:''fN

The result follows by letting LeIDIDa 3.9.5 Suppose mapping

(1iM(A)

(T

.J..

f : JRM

+0)-

•

O. -+]RN

is a Lipschitz function. Then the

is l£M -measumble. Proof: By the previous lemma, we can ignore sets of arbitrarily small measure, hence we may assume that A is compact. If A is compact, then

{y: 1i M -

N

(A n

r

1 (y)

~

t} =

nVj, j

here Vj is the open subset of JRN consisting of all points y for which An f-l(y) has a finite cover by open sets U; of diameter less than Iii such that

~T L.J i

M-N

(diam(U;») M 2

-N


+ !... J

•

3.9. THE GO-AREA FORMULA

141

Proof ot the Co-area Formula By Theorem 3.8.10 an~ (3.148), we may assume that D J(a) exists at every point a E A. First we prove the result under the additional assumption that JNJ(a) > 0 at every point of A. By LUBin's Theorem 1.3.12 we may assume that D J(a) is the restriction to A of a continuous function. By Egorofl"s Theorem 1.3.13 we may suppose that IJ(y) - J(a) - (DI(a),y - a)1 . I!I-al

converges uniformly to 0 as !I E A approaches a E A. It is plain that, for any £ > 0, conditions (i)-(Ui) of Lemma 3.9.3 are satisfied in any subset of A that has small enough diameter. Finally, to complete the proof, we need to consider the case in which IN J = 0 holds on all of A. In that case, the left-hand side of (3.144) is o. We need to show that the right-hand side of (3.144) also equals o. To this end, consider J. : lR,M+N -+ lR,N defined by . (x, !I) t-+ I(x)

+ EX.

We can apply what has already been proved to the .set A x [-I,I]N C ntM x ~.

We have .eM+N(A x [-1, I]N)

r

JAX[-l,ljN

= 2N .eM(A), JNJ.

JNJ.d.e M+ N = (

JRN

~

£ [£ + Lip(J)]N-l, and

ll M [(AX [-I,I]N) nJ;l(z)]d.eNz.

By (3.148) observe that G(M, N) llM [(A x

iN 1

?: =

[-1, I]N) n I.-l(z~

ll M - N [(A x [-1, I]N) n I.-l(z) n n-l(y)] d.cNy

[-l,ljN

llM-N[Anrl(Z-fy)]d.eNy. .

Thus 2N .eM (A) f [£ + Lip(J)]N-l

?:

(

JAX[-l,ljN

?:

1 G(M, N)

=

1 G(M, N)

=

G(:; N)

JNJ.d.c M+ N

r 1[-l,lIN llM-N[AnJ-l(z-£y)]d.cN!ld.cNz

JRN

1 iN

[-l,ljN

(llM-N[Anrl(z-£y)]d.cNzd.cNy

JRN

ll M- N [A n rl(z)] d.e N z

142

CHAPTER 3. MEASURES

holds, where the last equation holds by translation invariance. Letting E ../. 0, we see that

f 1lM - N [Anj-l(z)]dC N z=O. JRN

•

Chapter 4

Sobolev Spaces 4.1

Basic Definitions and Results

This book has given substantial attention to the C" spaces. Such spaces are suitable classes from which to select the defining function for a domain, and the C" spaces are' natural in a number of other geometnc contexts. However, in the study of partial differential equations and Fourier analysis, the Sobolev spaces are more convenient. The definition of the Sobolev spaces is less near the surface than that of the spaCes, but theorems about Sobolev spaces are more accessible. In the end, the Sobolev imbedding theorem allows one to pass back and forth between the C" spaces and the Sobolev spaces (however one must pay with a certain lack of precision that is present in the imbedding theorem). We will need the following basic definitions.

c"

Definition 4.1.1

(i) V is used to denote the space of infinitely differentiable functions on ]RN having compact support and topologized by uniform convefYence on compact sets. (ii) The Fourier transform of f E L1(JRN) is denoted by j and is defined by (4.1)

(iii) The Fourier Inversion Theorem (e.g. see Stein and Weiss /1j) states that if f, j E L1 (]RN), then f(x)

= [ j«(,)e27fiz·(dCN(, iRN

holds for almost every x E

]RN •

143

(4.2)

CHAPTER 4. SOBOLEV SPACES

144

For simplicity, we begin our discussion of Sobolev spaces on the domain RN, or all of space. We follow the discussion in Krantz [4). Definition 4.1.2 II tP

eV

and 8

eR

then toe define the nonn

(4.3) We let H-(lRN

)

be the cl08ure olV with respect to II • 11_.

In the case that

8

is a non-negative integer, then

Therefore

tP e H- if and only if if, . [EIQI~.I~IQl e L2. This last condition means that if,~Q e L2 for all multi-indices a with By the Plancherel theorem (see Stein and Weiss [1]), we have

(!)

Q

tP e j} Va such that

101 $

lal

$

8.

8.

Thus we have Proposition 4.1.3 118 is a non-negative integer, then

Here derivatives are interpreted in the sense

0/ distributions.

Notice that if s > r then H· C Hr because

The Sobolev spaces turn out to be easy to work with because they are modeled on L2-indeed each H· is canonically isomorphic as a Hilbert space to L2 (exercise). But they are important because they can be related to the more classical spaces of smooth functions. That is the content of the Sobolev Imbedding Theorem: Theorem 4.1.4 (Sobolev) Let 8> N/2. 1// e H·CIR.N), then I can be corrected, uniquely, on a 8et 01 measure zero to be continuous. More generally, il k e {a, 1,2, ...} and i/I e H·, 8 > N /2 + k, then I can be corrected on a set 01 measure zero to be C".

4.1. BASIC DEFINITIONS AND RESULTS

145

Proof: For the first part of the theorem, let I E H8. By definition, there exist 4>; E V such that 114>; - JIIH' -+ o. Then

114>; - JIIL2 = 114>; -

Iilo ~ 114>; -

III~ -+ O.

(4.4)

Our plan is to show that {4>;} is an equibounded, equicontinuous family of functions. Then the Ascoli-Arzela theorem (e.g. see Rudin [1]) will imply that there is a subsequence converging uniformly on compact sets to a (continuous) function g. But (4.4) guarantees that a subsequence of this subsequence converges pointwise to the function f. So f = 9 almost everywhere and the required assertion follows. To see that {4>j} is equibounded, we calculate that

14>; (x) I =

~

II

e- 21fiz • (~;{{) c.l{1

I 1~;({)I(1

+ 1{12)B/2(1 + 1{1 2)-B/2 c.l{

~ (/1~j({)12{1+1{12)Bc.l{r/2. (/{1+ 1{12)-Bc.l{r/2 Using polar coordinates, we may see easily that, for s > N/2,

1(1

+ 1{1 2)-B c.l{ <

00.

Therefore

14>; {x)1 ~ CII4>;IIH- ~ C' and {4>;} is equibounded. To see that {4>j} is equicontinuous, we write

14>;{x) - 4>; {y)1 =

II ~;({)

(e- 21fiZ :(

-

e- 2,,-i ll ·() d{1·

Observe that le- 21fiZ • ( - e- 2,,-ill • (I

~ 2 and, by the mean value theorem, 21fiz 2 le• ( _ e- ,,-ill· (I ~ 211"Ix - yll{l·

Then, for any 0

< f < 1, le- 21riZ ·( _ ~

211"< Ix

e-21rill·(ll-'le-21riz.~

_

e-21rill·~I'

- YI
Therefore

14>;{x) - 4>; (y)1

< C/

1~;{{)lIx -

~ Clx -

~

YI'

ylfl{I' c.l{

I 1~;({)I(1

+ 1{12)'/2

d{

Clx - YI'II4>; IIH- (/ (1 + 1{1 2)-B+<

d{) 1/2 ,

CHAPTER 4. SOBOLEV SPACES

146

where the last inequality is a consequence of the Schwarz inequality. H we select 0 < E < 1 such that -8 + E < -N/2 then We find that

!

(1 + leI 2)-s+f

d{

is finite. It follows that the sequence {tPi} is eciuicontinuous and we are

done. The second assertion of the theorem may be derived from the first by a • simple inductive argUment. We leave the details as an exercise. Remark 4.1.5 8 = N/2, then the first part of th~ ~heorem is false (exercise). In fact the sharp statement that one make is that in this, case,

(i) H

can

I lies in the space of functions of bounded mean oscillation or that I is exponentially integrable (see Stein [1], Folland and Stein [1], Adams [1]).

(ii) Theorem 4.1.4 may be interpreted as saying that HS C C" for 8 > k + N /2. In other words, the identity provides a continuous imbedding of HB into C". A converse is also true. Namely, if HB c C" for some non-negative integer k, then 8 > k +.N/2. . . To see this, notice that the hypotheses Uj -+ U in HB and Uj -+ v in C" imply that U = v. Therefore the inclusion of H" into C" is a closed map. It is therefore continuous by the closed graph theorem. Thus there is a constant C such that (4.5) Now, for x E JRN fixed and a a multi-index with lOll ::5 k, the tempered distribution e~ defined by

e~(tP) = (::0) t/J(x) is bounded in (C")· with bound independent of x and a (but depending on k). (See Stein and Weiss [1] or Krantz [4] for a discussion of tempered distributions.) Hence, by (4.5), {e~} forms a bounded set in (Ha). == H-II. As a result, for lOll :5 k we have that

lIe~ IIH-' = =

:5

(! 1(e:) 12 (1 + leI2)-e d{) 1/2 (! 1(-27rie)Oe2"';"'·(12 (1 + leI2)-s~) 1/2 C (J (1 + leI 2)-e+ d{) (e)

10 1

1/2

4.1. BASIC DEFINITIONS AND RESULTS

147

is finite, independent of x and a. But this can only happen if2(k-s) -N, that is if 8> k + N/2.

<

We cannot proceed any further without giving a precise definition of Sobolev spaces on a domain. In fact we shall give three such definitions, and in the next section we will compare and contrast them. Definition 4.1.6

(i) Let 0

c

lllN be any domain. Let s ~ 0 be an.integer. Define the norm

1If1IH'(O) =

E 11:~112

loiS-

L (0)

.

Let 'PB consist of those f e C 8 (11) for which IIfIlH'(o) is finite. This is clearly a linear space. Define HB(O) to be the closure of pB in the II • IIH'(o) norm. (ii) Let 0, II • IIH'(o) be as in the last definition. Let W8(O) consist of those f e L2(11) such that (aO /axQ)! e L2 for all lal ~ s. Here derivatives are, perforce,. interpreted in the weak sense. (iii) Let 0, II • IIH'(o) be as in the preceding two definitions. We define W3(O) to be the closure in the II • IIH' topology of the space 9~(11). Remark 4.1.7 (i) It is clear from the definitions that W3 c WB and W3(O) C HB(I1). That these inclusions are proper when s is large enough is clear from the Sobolev imbedding theorem. For if s > N /2 then an element of W3(O) would necessarily vanish on al1. And density in the II • IIH'(o) nonn would imply uniform density. Thus W3(O) could not be dense in either of the first two spaces. However more is true. In fact if s> 1/2, then W3(O) is not dense in WB(O), nor is it dense in HB(I1). The reader may try his hand at this as an exercise, or consult Taylor

[1]. (ii) It is a striking fact, discovered as recently as 1964 (see Meyers and Serrin [1]) that, for any domain 11 C lllN, H8(O) = WB(I1). The proof of this statement is a standard, but tricky, adaptation of the usual approximation by cutting off and convolving with a family of Friedrichs mollifiers. We omit the details, but refer the reader to

Adams [1]. Exercise: Imitate the proof of the Sobolev bnbedding Theorem 4.1.4 to prove Rellich's lemma: H s > r and if O2 CC 111 then the inclusion map

given by

I

t---+

1102

is a compact operator (see Krantz [4] for the details).

4.2

Restriction and Trace Theorellls for Sobolev Spaces

In the preceding section, we discussed the most basic facts about the Sobolev spaces. In this section, we will adapt some of our earlier results to bounded domains iIi space. First, we discuss the significance of the so-called "restriction theorem:" Let S = {(Xl, ... ,XN-I,O)} C,RN. Then S is a hypersurface, and is the boundary of {(XI, ... ,XN) E,RN : XN > OJ. It is the simplest example of the type of geometric object that arises as the boundary of a domain. It is natural to want to be able to restrict a function f defined on a neighborhood of S, or on one side of S, to S. If I is continuous on a neighborhood of S, then the restriction of I to S is trivially and unambiguously defined, simply because a continuous function is well-defined at every point. If, instead of being continuous, I is an element of the Sobolev space H N /2+< , then we may apply the Sobolev imbedding theorem to correct I on a set of measure zero (in a unique manner) to obtain a continuous function . . The corrected I may then be restricted to S. By contrast, restriction prior to the correction on the set of measure zero is prima lacie ambiguous. This is because a Sobolev space "function" is really an equivalence class of functions any pair of which need only agree up to a set of measure zero. The set S itself has measure zero, and thus two different elements of the equivalence class of I may have different restrictions to S. We wish to develop a notion of calculating the restriction or "trace" of a Sobolev space function on a hypersurface that applies to Hr for r < < N /2 and is such that the restriction operation works naturally in the context of Sobolev classes without relying on the Sobolev imbedding theorem. Our first result is a bound on norms. This bound, given in estimate (4.6), shows us when the trace of a Sobolev function is defined, thus the theorem is known as the Sobolev trace theorem. Theorem. 4.2.1 Identify S with ,RN-I in the natural way. Let s

>

1/2.

Then the mapping from C.;"'(,RN} to C.;"'(JRN-l} defined by 4> t---+ 4> 1s

extends to a bounded linear operator from H·caN ) to H S - I / 2 (,RN-l). That is, there exists a constant C = C(s} > 0 such that (4.6)

Remark 4.2.2

. (i) Since H·(R.N) and H·-l/2(R.N-l) are defined to be the closures of ~(JlN) and O~(R.N-l) respectively, we may use the theorem to conclude the following: H T is any {N -1}- dimensional affine subspace ofJlN, then a function f E H8{JiN} has a well-defined trace in H·-l/2 on T, provided that 8 > 1/2. Conversely, we shall see that if 9 E H8-1/2(T},8> 1/2, then there is a function 9 E HS(RN) such that 9 has trace 9 on T.

(li) It is a bit awkward to state the theorem as we have (that is, as an a priori estimate on O~ functions). As an exercise, the reader should attempt to reformulate the theorem directly in terms of the H8 spaces to see that in fact the statement of Theorem 4.2.1 is as simple as it can be made.

Proof of TheoreDl 4.2.1: We introduce the notation (X',XN) = (Xl, ... ,XN)

for an element of]RN. HuE Cg"(JRN), then we will ~se the notation ur(x') to denote u(x', 0). Now we have

111.Irll~'_1/2(RJV_l)

2 1fR N-l IUr(e') 12(1 + le'1 },-1/2 d{'

=

=.f [- foo {}{} IUI(e',XN}1 2(1 + le'1 2y-I/2 dxN ] .1RN-l 10 XN

df.'. (4.7)

Here UI denotes the partial Fourier transform in the variable x'. The product rule yields that, if D is a first derivative, then D(lhI 2 ) $ 21hl . IDhl. Therefore (4.7) does not exceed -

foo IUI(e', xN}1

2 {

1RN-l10

0/ 2

Since 20/{j $

$

·1 {}{}

XN

UI(e', XN}

I· (I + le'1 2},-1/2

dxN df.'.

+ {32, the last line is f

foo

1RN-l 10

I~UI(e',xN}12 dxN(1 + le'1 2},-1 df.' {}XN

foo IUI(e', XN }1 2 dxN(1 + le'1 2 }, d{'

+ f

1RN-l10

_

1+11.

Now apply Plancherel's theorem to term II in the XN variable. The result is II

~

0

~

Ollull~'(RN).

f f 1u(e',eN}1 2 df.N(I+le'1 2 )'d{' 1RN-l1R

150

CHAPTER 4. SOBOLEV SPACES

Planchere1's theorem, applied in the XN variable to the term I, yields

I

~

C {

~

C {

~

C

=

CliuIlH'(RN).

2 . {Iaa ii1(e"XN)1 dxN(1 + le'1 2 )B-l df.' JRN-IJ R ZN

JRN-l

2 2 2 J{R lu(e', eN )1 ·leNl deN (1 + le'1 )B-l df.'

kN lu(e)1

2 •. (1

+ lel 2 )B de •

As an immediate corollary we have: Corollary 4.2.3 Let u e HB(JRN) and let Q be a mul~i-inde:z: such that s> 101 + 1/2. Then DQu has trace in H B -l ol-l/2(RN-i). The reader will note that, in the trace theorem, it is only necessary to have u e HB+1/2(JR~) in order to obtain a well-defined restriction to JRN-l. In other words, the function .to be restricted need only be defined on one side of the boundary. Now we present a converse to the Sobolev trace theOrem. This converse is the extension theorem for Sobolev spaces. We continue to use the notation S = {(z', 0) e RN}. In this theorem it is convenient to let DN denote· i(8/8zN). TheoreDl 4.2.4 Let k be a non-negative integer, and suppose s > 1/2 + k. If if>o, •. . ,
IIfIlH'(RN) ::; C •. 1t

L lI¢ljll~'-H/2(RN-l).

j=O

Proof: Let h We define

e

C~(R),

h

and

fez) =

== 1 in a neighborhood of the origin, 0 ::; h::;

f

e- 27ris '

•

1.

fii1({',XN) de'.

This is the function f that we seek. For if m is any integer, 0 ::; m ::; k, then

D~f(z',O)

=

f

e- 27ris '

'('

D~ii1(X',XN)I"'N=o~'

4.2. SOBOLEV SPACE RESTRICTION AND TRACE THEOREMS

=

JeX

2 ,..iz'

0 ('

im

t ~, E(m) ~(-iXN)j I j=O ,. l=O t

J

e -2,..iz'

0 ('

~ .!.-, ,·m L.J

. J.

8:-~; [h(XN(l + le'12)1/2)]

J

aN=O

. ~(e') df.'

(m) .'(_ .);

j=O ,. ,

8xN

=

ZN=O

8:i:'N

8:-~l [h(XN(l + IfI2)1/2)]1

8x N

=

151

t

IaN=O .~(e') df.'

e- 2 ,..iz' 0(' ~(e') df.'

= c,t;m(X'), verifying our first assertion. Now we need to check that f has the right Sobolev norm. We have

IIfll~2

=

kN li(eW(l + le1 2

=

kNI(Ui(e',.»)~(eN)12(1+leI2Yd{

)6 dS

Using the fact that 12:;=0 (j12 .$ C· (2:;=01(;1 2 ) , where the constant"C depends only on k, we see that (4.8) does not exceed

Now write df.

= df.' df.N. We make the change of variables eN t-+ eN(l + le'1 2)1/2.

152

CHAPTER 4. SOBOLEV SPACES

Noting that

1 + lel 2 = 1 + le'I 2 + leNI 2

1 + le'I 2 + leNl2(1 + le'1 2) (1 + le'1 2)(1 + leNI 2),

1-+

=

we see that (4.9) does not exceed

C

"

~ 1 r (IA(j)( )12 1 1'12)1/2 f;;t, 0!)2 JRN-l JR h £.N (1 + le'1 2)Hl (1 + e xl~(e')12 [(1 + 1£.'1 2)(1 + leNI2)]8 de' deN

=

"

r

C ~ _.1_ (]!)2 JRN-l

f;;t,

x

"

::;

C];

::;

C•."

l~j(£.'W(1 + le'1 2)S-j-l/2 de'

Llh

W (eN)1 2(1

0~)211t/>j IIH'-i-

+ leN12)BdeN .

1/

2(lR N- 1)

•

IIhll~'+k(R)

L" lIt/>jll~'-i-l/2(RN-l)'

j=O .

•

completing the proof of our extension theorem. Remark 4.2.5

(i) The extension Ui was constructed by a scheme based on ideas that go back at least to A. P. Calderon-see Stein [1] and references therein. (ii) Consider a smoothly bounded domain n c JRN. Let {l!j}~=1 be a covering of an with open subsets of JRN. We may choose these open sets in such a way that there are diffeomorphisms ~j : Uj -t Wj C JRN such that ~i(aO) = Wi n {x E JRN : XN = OJ. This claim follows from the implicit function theorem. We may assume that there is a universal finite constant C such that IIV~jIl8uP ::; C and IIV[~j]-lllsup ::; C for each j. Let,pj be a partition of unity subordinate to the covering {Uj} of an such that Ei,pi := 1 in a neighborhood of an in JRN. Suppose that f E H8(O). Fix an index j, 1 ::; j ::; k. We write ~j(n n Uj) := V;. Then 9j == ,pi . f 0 (~j lies in H8(V;). According to our trace theorem for Sobolev functions on the upper half space ~, the function 9i has a trace on ~j(an). Unraveling all the notation, we find that ,pj . f has a trace on an. Summing over j, we conclude that f itself has a trace on an. We shall not examine

llvi)

4.3. EXTENSION THEOREMS FOR SOBOLEV SPACES

153

minimal hypotheses on the smoothness of 80 that will guarantee the validity of the trace theorem for elements of HB, 8> 1/2. Clearly it suffices to aSsume that the boundary is Cle with k > 8. Remarks similar to those in the preceding two paragraphs apply to show that the extension theorem has a valid fonnulation for Sobolev functions on the boundary of a domain with sufficiently smooth boundary. We shall say no more about the matter here.

4.3

Domain Extension Theorems for Sobolev Spaces

Next we turn our attention to domain extension theorems for Sobolev functions. Such theorems are related to the converse of the trace theorem, but they have a slightly different formulation. We will be concerned here with extending a Sobolev function from a given domain in JRN to all of space. It is easy to see that a successful extension theorem will necessitate some regularity condition on the boundary of the domain in question. For let us consider the situation in JR2 • Fix an integer k > O. Let

Define f(xI, X2) = I/Xl. If k 2: 2 then f E L2(Ok). If k 2: 4 then f E Wl(OIc). More generally, if k 2: 28 + 2 then f E W8(Ok). But for s > 1 we cannot expect f to extend to a function in W 8 (JR2). If it did, then the extended function would be (after correctioE. on a set of measure zero) continuous-by the Sobolev theorem. Since f is unbounded at (0,0), that conclusion is clearly impossible. Very sharp conditions on the boundary of a domain in Euclidean space, in order to guarantee a valid extension theorem, are known (see Jones [1]). We shall not attempt such a precise discussion, but shall instead treat some of the standard results. Definition 4.3.1

(i) Let 0 C JRN be a domain. A linear operator

is said to be a simple m-extension operator for 0 if 1. Eu(x) = u(x) for almost every point x E OJ 2. IIEullw'"(RN) $ K ·lIullw'"(o).

CHAPTER 4. SOBOLEV SPACES

154 (ii) With 0 as above, we say that

E : {functions on O} --+ {functions on RN}

is a total extension operator il the restriction 01 E to wm(o) is a simple m-eztension operator lor each non-negative integer m. We shall present three extension theorems for Sobolev spaces (and this three part unity is quite typical of extension questions for other function spaces too). In our discussion, we shall restrict attention to hounded domains in Euclidean space. This choice makes the statements of the theorems a bit cleaner. A treatment of unbounded domains may be found in Adams [1] and in Stein [1]. It is also possible to develop Soholev space theory for domains in a manifold-see Hormander [1]. .' . Theorem 4.3.2 II 0 C]RN is a bounded domain with cm boundary, then there exists a simple m-extension operator lor non-negative integer m.

each

Proof: First let us assume that 0 is the upper half space ~

= {(X17X2, ••• ,XN) E aN : XN > OJ.

(Of course this domain is not bounded, but that will be seen not to matter in what follows.) It is convenient to denote a point of 1R~ by (X',XN), where x' is a shorthand for (Xl, ••• ,XN-t). Set u(x) if XN > 0 Eu(x) = { m+1 ,. • Lj=l ~ju(x, -JXN) 1f XN :5 0,

where Ah ... , Am+1 are the unique solutions to the m+ 1 equations in m+ 1 unknowns given by m+1

L: (_j)A: Aj = 1

k

= 0,1, .. . ,m.

;=1 Also define the auxiliary operators

E u(x) _ { u () x a -

if

r::;::-+/ (- j)a

N

Aju(x', - ixN)

XN

>0

if xN:5 0,

for a = (ah ... ,aN) any multi-index with lal :5 m. IT u E C m (~) then it is straightforward to calculate that

Eu

E

Cm(]RN),

lal:5m,

4.3. EXTENSION THEOREMS FOR SOBOLEV SPACES {

ID'"Eu(xWdx=

JRN·

(

JR~

155

ID'"u(xWdx

density result Remaark 4.I.5(ii», i.e., Hm = This completes the proof of the theorem when the ·domain is the half space. IT now n is a bounded domain with em boundary, then we follow a by now familiar procedure of covering an with finitely many open sets Uj, each diffeomorphic to a ball, and each equipped with a mapping i)j : Uj ~ Wj C ]RN such that i)(Uj nan) = Wj n {x E ]RN : XN = O} and i)j(Uj n n) = Wj n ~. This, in effect, reduces the problem on n to that already treated on the upper half space. To wit, if I E wm(n) then one extends each of the functions 10 i)j -1 from Wj n ~ to wj, then pulls the extension back via i)j and patches all these together. Details are left to the reader. • By the previously

DOted

wm, these inequalities extend to all of W m.

A refinement of the proof of the last theorem yieids the following stronger result. We omit the proof, and refer the reader to Adams [I] for details. Theorem 4.3.3 If (l c aN is a bounded domain with there is a total extension operator lor n.

em

boundary, then

Using deeper techniques, related to the Cafder6n-Zygmund singular integral theory (for which see Stein [1]), it is ac~ually possible to prove the following still stronger result. Again we omit the proof. Proofs may be found in Adams [I] and Stein [I]. As previously noted, an even sharper result appears in Jones [I]. The result of Jones is precisely sharp in dimension two. Theorem 4.3.4 If n c aN is a bounded domain with Lipschitz boundary, then there is a total extension operator for n.

Chapter 5

Smooth Mappings 5.1

Sard's Theorem

Sard's Theorem is an important tool in differential topology and in Morse theory. We shall begin our discussion of Sard's theorem by treating the most elementary form of the result, and then shall later develop generalizations and variants of the theorem. It is worth noting that the basic idea here was discovered by A. B. Brown in 1935 (see Brown [1]). Later, in 1939 and 1942, the result was rediscovered by A. P. Morse (see Morse [1]) and A. Sard (see Sard [1]). The method of proof of Sard's theorem is quite robust and can be modified to yield a number· of interesting results. We shall close the section with a discussion of one of these variants that is known as "hard Sard." ]iN, V C:: JRM be open sets and let f : U -+ V be a C 1 function. We say that y E V is a critical value of f if there is "an x E U with f(x) = y and V' f(x) has rank less than M. The point x is of course called a critical point.

Definition 5.1.1 Let U C

We begin with the following special case of the theorem of Brown, Morse, and Sard: TheorelU 5.1.2 Let U c ]iN, V C]RM be open sets and let f : U -+ V be a C 2 function. Assume that 0 < N < 2M. Let C denote those points in U where V' f = O. Then the image of C under f has Lebesgue tJolume measure

equal to zero. Proof: For each N-tuple of integers k

= (k1 , ..• , kN) let

= {X=(Xb ... ,XN) E]iN :"kj $Xj $kj+1,j=1, ... ,N}. Then ]iN = UIcQIc and the QIc have disjoint interiors. Set QIc

Sic = {y E V: 11 E fCC) and 3x E QIc such that f(x) = y}.

157

CHAPTER 5. SMOOTH MAPPINGS

158

Then U"S" equals I(c), and it is enough for us to show that each S" has rp.e~ zero. Given this reduction, we may as well suppose that I has as its domain the cube Q = {z e ItN : O:oS zi:oS 1,; = 1, ... ,N}. Now the function V I is uniformly continuous on Q. Better yet, V 2 I is continuous on Q, and hence is bounded. It follows from the mean value theorem that V I itself is Lipschitz. Let K be the Lipschitz constant:

IV fez) - V I(z + h)1 :oS Klhl·

(5.1)

Let E > O. Choose a 6 > 0 such that if z, t E Q and Ix - tl < N5, then IV/(z)-V/(t)1 < E. In fact 6 = E/(KN) will do. We may certainly assume that 5 < E. Select an integerp, (1/6)+1 ~ p > 1/5. Note that 1/5,..., (KN)/E. Break Q up into rP closed subcubes, with side length equal to lip, sides parallel to the axes, and disjoint interiors. Consider anyone of the ~ sub cubes, call it P, that contains a critical point. The diameter of this cube is less than .JN5. And there is a point in the cube where V I vanishes. It follows that IV fl < E on the entire cube. We conclude, from the mean value theorem applied to I restricted to one dimensional segments in P (for instance), that the image of the cube P under I has diameter at most E·.JN5 < E· .IN/p. Thus it has volume at most (5.2)

Summing over all subcubes P that contain critical points (there are at most fCC) has volume at most

rP of these) we find that the set pN.

[2v'NE/P]M =

(2.JNE)M

.pN~M.

(5.3)

But now recall that p"" KN/E. Thus (5.3) is essentially

(2vNf)M (KN)N-MEM-N :oS (2.JN)M (KN)N-M 2M-N. f

Since E > 0 was arbitrary, and since 2M - N set I(C) has measure zero.

> 0, we conclude that the •

In fact Sard's theorem is true with no restrictions on the dimension of the domain and of the range, and it is true for the set of all critical values-not just the image of the points where the gradient vanishes. However a certain smoothness of the function f is necessary; examples that demonstrate the necessity of the smoothness requirements may be found in Federer [4], p. 317 JJ. The complete statement of the classical Sard theorem is this: Theorem 5.1.3 Let U C ]RN, V C IRM be open sets and let I : U -+ V be a C" function, where k ~ N/M. Then the set 01 critical values of I has Lebesgue M -dimensional volume measure equal to zero.

5.1. SARD'S THEOREM

159

Proof: In this proof we shall assume that the map is Coo, and we shall not give the more delicate proof that is valid for Cle functions, k ~ N / M. We refer the reader to Federer [4], p. 316 for a full account of these matters. The proof that we present comes from Milnor [1] and Sternberg [1]. We proceed by induction on N. Notice that the statement of the theorem makes sense, and is obviously true, for N = o. That is the starting point of our induction. Assume now that the result has been proved for dimension N - 1. Let C be the set of points where the rank of V f is less than M. Let C1 C C be the set of points where V f = 0 (this set was denoted by C in Theorem 5.1.2). More geneJ;ally, let C i denote the set of points where V f = 0, V 2 f = 0, ... Vi f = o. Of course each C i is closed and we have

We shall prove the following three assertions: 1. The image of C \ C1 under f has measure zero. 2. The image of Ci \

CHI

3. The Jmage of Cj under

under f has measure zero, i = 1,2, ....

f has measure zero if j is sufficiently large.

Step 1: If M = 1 then C = C 1 and there is nothing to prove. So assume that M ~ 2. Choose t E C \ C 1 • By definition, th~re is some partial derivative 8ftl8xj(t) that does not vanish. After juggling indices, we may assume that 8fd8xl(t) I: o. Define h(x) =·(ft(X),X2' ... '_~N).

Then the inverse function theorem implies that there i~ a neighborhood V of t on which h is a diffeomorphism onto an open set V' C RN. Now the composition 9 = f 0 h- 1 will map V' into RM, and the set C' of critical points of 9 is nothing other than h(V n C). In other words, the set g(C') of critical values of 9 is equal to f(V n C). So it is enough for us to determine the measure of g(C'). For a point of the form (S,X2, ... ,XN) E V', observe that the image g(S,X2, ... ,XM) belongs to the hyperplane {s} x RM-l. Therefore 9 maps hyperplanes into hyperplanes. Now define g. : [{s} x lIlN -

1]

n V' -+ {s}

to be the restriction of 9 to the hyperplane {s} Jacobian matrix of 9 has the form

(!

0 ) (8gf/8 . X t)

X

X

lIlM -

1

lIlN -1. Notice that the

160

CHAPTER 5. SMOOTH MAPPINGS

Thus a: point of {8} X jRN-1 is critical for g8 if and only if it is critical for the map 9 itself. By the inductive hypothesis, the set of critical values of g8 has M - 1 dimensional measure zero. By Fubini's theorem (taking the union over s), it follows that the set of critical values of 9 itself has M dimensional measure zero.

Step 2: Fix j, and consider a point t E C j \ Cj +1. By definition, some U + 1}1\ derivative of some component of 1 is not zero at t. Thus there is some ktb order partial derivative D, and index t, and another index m, such that w(t) == Dft(t} = 0 but 8/8x m Dft(t) ::F O. We may take m = 1. Then the map h : U -+ jRN defined by

is a diffeomorphism of some neighborhood V of t onto an open set V' C JR.N. The map h maps C j n V into the hyperplane {OJ x lIlN - 1 • Proceeding as in Step 1, we consider

We let" g* :

[{O}

X

jRN-l] n V'

-+

JR.M

denote "the restriction of 9 to the hyperplane {OJ x lIlN - 1 • By induction, the set of critical values of g* has (M - 1) dimensional measure zero. But each point of h( C j n V) is a critical point of g* (because aU deriVatives of order not exceeding j vanish by definition). Thus [g*

0

h](Cj

n V)

= I(C n V) j

has measure zero. Since we may cover Cj \ C j +1 by countably many such sets V, we may conclude that I(Cj \ Cj+1) has measure zero. Step 3: Refer to the proof of Theorem 5.1.2 above. For a point x E Cj we may replace line (5.1) in that proof by I/(x

+ h) -

l(x)1 ~

K'lhl j -

1•

Thus the estimate for the volume of the image of a cube in line (5.2) becomes

Since 6 < t, in the final calculation the term t 2M - N is replaced by t jM H j > N/M then the rest of the proof goes through. •

N •

5.1. SARD'S THEOREM

161

Implicit in the proof that we have just presented of Sard's theorem is the following remarkable lemma of A. P. Morse. We record it here for interest's sake, and invite the reader to g;.ve· a proof (details may also be found in Sternberg [1], p. 49). LeDlDlB 5.1.4 (Morse) Let A c RN and let q ~ 0 be an integer. Then we may decompose A fI6 A U~Ai such that: Let U be a neighborhood 0/ A and / : U -+ RM a C' mapping. Suppose that every point 0/ A is a critical point all. Then there are functions ei(f) (these junctions depend on f) such that each e, satisfies ei(f)·~ 0 monotonically fI6 f decreases to o and

=

lor any x, 11 E Ai. Remark 5.1.5 We note explicitly that the sets Ai in the decomposition of A in Lemma 5.1.4 are independent of the choice of the function I. Here is a typical application.of Sard's theorem. Example 5.1.6 Let I: ]RN ~ ]R be a C lc function, k ~ N, that is proper, where "proper" means that the inverse image of any compact set is compact. Then for almost every c E ]R the set Se

== {x

E]RN :

f(x) =c}

is either empty or is a C Ic compact manifold. To see this, notice that Sard's theorem guarantees that the set C of critical values for I has measure zero. If c is in the complement of C, and is in the image of I, then every point of f-l(cjis not a critical point. Thus V I "I. 0 in a neighborhood of each such point, and therefore 1-1 (c) is a CIc hypersurface in a neighborhood of each such point. We have established that if c ¢ C and if f-l(C) is nonempty, then it is a CIc manifold. The properness of f guarantees that it is compact. Now we give a formulation of a version of "hard Sard." We shall only sketch the proof, and refer the reader to Hirsch [1] or Federer [4] for a more detailed treatment.

Theorem 5.1.7 Let U C]RN, V C ]RM be open sets and let I: U -+ V be a C lc junction. Let us say that a critical value w is 01 rank l i/ w = lex) lor at least one point x at which the Jacobian matrix has rank not exceeding l. Denote by Ct the set 0/ points in V that are critical values of rank l. Then 'H.l+(N-l)/Ic(Cl)

In other words, the set 0/ critical l1alues not exceeding l + (N -l) / k.

= O.

0/ rank l has Hausdorff dimension

J.U":'

UHAP'l'ER 5. SMOOTH MAPPINGS

The spirit of the proof is that, near a point at which the Jacobian of / has rank l, the mapping will collapse a small N-dimensional ball in U to an essentially l- dimensional ball in the image. Since the latter balls cover the set Cl, it is clear that this puts a bound on the Hausdorff dimension of

.

~.

5.2

Extension Theorems

We have already (in Section 1.1) defined what it means for a function / : U -+ ]RM to be C" when U is an open subset of ]RN. A difficult question arises when / is not a priori defined on an open set. What should it mean in that case for / to be C"? An easy definition, which is very often used, is to require the existence of an extension of / to a larger open domain that contains the original domain and to demand that the extension be C" on that larger domain. While this definition is expedient, it really begs the question. One should be able to determine from information about / on 'its given domain whether or not / is in a particular smoothness class. It should not be necessary to appeal to an extension "oracle" to learn the answer. The replacement for an extension "oracle" is an extension theorem. The following classical extension theorem is well-known and can be found in many references (e.g. see Dugundji [1]). Theorem 5.2.1 (Tietze's Extension Theorem) Let S be a closed subset 0/ RN. A function / : S -+ JRM is continuous i/ and only if / has a continuous extension to all 0/ JRN . Tietze's Extension Theorem is a model for what we would like to accomplish: A characterization of the functions that are restrictions of those in a certain category. The preceding extension theorem is more topological than analytical, requiring little from the geometry of JRN • To move into the arena of analysis we deal first with Lipschitz functions. Recall from Definition 3.1.15 that / : X -+ Y, with X and Y subsets of Euclidean spaces, is a Lipschitz function if there exists C < 00 such that

and the smallest such C is called the Lipschitz constant of /. The following theorem is known as Kirzbraun's Extension Theorem. Theorem 5.2.2 (Kirzbraun [1]) Suppose S C ]RN. A function / : S -+ RM is Lipschitz if and only if / has a Lipschitz extension to all ofntN. The extended function may be taken to have the same Lipschitz constant. Proof: It is clear that the restriction of a Lipschitz function is Lipschitz. The heart of the matter is proving the existence of an extension. Let us

5.2. EXTENSION THEOREMS

use C to denote the Lipschitz constant for f: S-+ aM. In case M can simply use the followiilg fonnula, due to H. Whitney, to set F(x)

= sup{J(z) -

Cdist(z,z) : z E S},

= 1, we (5.5)

and easily check that this defines an extension of / with the same Lipschitz constant. In case M > 1, the fonnula (5.5) can be applied to each component to obtain a Lipschitz function, but the Lipschitz constant is generally larger than C. To obtain a Lipschitz extension having the same Lipschitz constant C requires a more elaborate construction. Consider the family :F of Lipschitz extensio~ of / (to some set T with SeT) which also have the Lipschitz constant C. This family is non-empty because·it at least contains the original function /. We define a partial ordering on :F as follows: Suppose 91 : T1 -+ aM and 92 : T2 -+ aM are both elements of :F, then we write 91 ~ 92 if and only if 92 is an extension of 917 i.e. S C T1 C T2 and 91(X) = 92(X), for all x E T 1 • (The same partial ordering is defined if we recall that a function from a subset of aN to lllM is a set of elements of lllN x lllM, and we partially order :F by inclusion.) By Hausdorff's Maximal Principle, :F has a maximal totally ordered sub-family !t. Let S be the union of the domains of the functions in !t. We define a function F : S -+ lllM by setting F(x) = 9(X) where 9 E !t and z E dom(g). One readily sees that Fis Lipschitz and that Lip(F) = C. We claim that S = lllN. IT not, then we can· fix Xo E lllN \ S. A contradiction will be reached if we can show that there is Yo E lllM such that

Iy -

whenever 11 = l"(x)

Yol :::; Clx - xol

and

xES,

that is, if we can show

o=F

n B(F(z),Clz - xol).

(5.6)

zes Since (5.6) involves an intersection of compact sets, it suffices to show that any such finite intersection is non-empty. Accordingly, let Xl, X2, ... ,X n E S be fixed. Set Yi = F(Xi)' Tj = IXi - %01, for i = 1,2, ... , n, and r* = sup{r1,r2, ... ,rn}. We know that for any sufficiently large value of 'Y n

K.., ==

nB(Yi, "Y7"i) ¥ 0. i=l

Set 'Yo

= inf{-y : K.., =F 0}.

Since (5.7)

164

CHAPTER 5. SMOOTH MAPPINGS

and the intersection of any finitely many of the sets on the right-hand side of (5.7) is non-empty, we see that

K'YO =/:0. It will suffice to show 'Yo S C. We may, of course, assume 'Yo > O. Note that K..,o must contain exactly one point, for if y', y" e K..,o, then we can use the following convexity argument:

I(y' + y")/2 - y,1 2

= Itl + 1/"1 2 /4 + ly,1 2 - (y' + y") • y, = I1l12/2 + ly"1 2 /2 - Iy' - y"1 2 /4 + ly,1 2 -y' . y, - y" • Yi

=

(ly' - y.1 2 + lIy" 2 2

S 'Yor, -

Iy' - y"1 2

2 _

'Yo

/2 - Iy' - y"1 2 /4

2

4(r*)2 ri'

holds for i = 1,2, ... , n, and (1/' + y")/2 'Y=

Yil 2 )

e

K.., with

Iy' - y"1 2 4(r*)2

< 'Yo,

contradicting the definition of 'Yo. By translating the coordinate system if necessary, we may assume to} = K..,o' Consequently, we have ly.1 S 'YOTiJ for i = 1,2, ... , n. We now claim that 0 is in the convex hull of {Yi : ly.1 = 'YoT'}' Were that not the case, there would exist an M - 1 dimensional plane separating the origin from {y. : IYil = 'YOTi}, but then for all sufficiently small E > 0 we would have B(O, E) on the opposite side ofthe plane from {Yi :"IYil = 'YOTi}, again contradicting the definition of 'Yo. Thus we can choose non-negative A1, A2, ... , An with and n

1 =

LAi, i=1 n

0

= LAW•. .=1

It follows that

5.2. EXTENSION THEOREMS

165

n

=

2

E

Ai Aj1li '1Ij

i,j=1 n

= E

AiAj [l1Ii1 2 + 11Ijl2 -11Ii -1IiP]

i,j=1 n

E

~

AiAj h~r1

+ 'Y~r~ -

~Ixi - xjl2]

i,j=1 n

= E

AiAj [2(Xi - xo)· 'Yo(Xj - xo)

i,j=1

=

I

n Ai (Xi - XO) 2 'Yo ~

+ h~ -

C 2)lxi - xjl2]

r

n AiAi IXi + (~- C 2 ) .J;1

xjl2.

(5.8)

If there were but one non-zero Ai, then the second term in (5.8) would vanish and the first would be positive, a contradiction. Thus there are at least two non-zero Ai'S and the second term in (5.8) must be non-positive, • forcing 'Yo $ C, as desired.

To discuss the extension of smoother functions, we first need to assess what the characterizing condition might be. To this end we consider Taylor's Theorem. It is elementary to see that, for a real-value~ function
~!

11

[
= _ ..!..¢(k) (0) k!

+

1 (k - 1)!

t [¢(k-1~~0) _

10

¢(k-1) (t)] d(l _ t)k-1

must hold. Induction leads to

~!

11

[¢(k) (0) - ¢(k) (t)] d(l _ t)k

(1 E ~¢(j)(0) + 10 [¢'(O) j=2 }. k

= -

¢'(t)] d(l - t)

0

k

= - E ~¢(j)(0) -

.p'(O) - ¢(O) +
j=2}·

which (after rearranging the terms) is the standard Taylor formula with remainder. Now suppose U is a domain in ]RN that contains the points a and x and the line segment connecting them, and suppose f : U -+ ]R is C k on an open set containing that line segment. Define the real-valued

CHAPTER 5. SMOOTH MAPPINGS

166

function of one variable, 4>, by setting

It is easy to see that

for j = 0,1, ... ,k. Thus we obtain Theorelll 5.2.3 (Taylor's Theorelll) If /: U -t 1R is C le on an open set containing the closed line segment joining a and x, then Ie

/(x)

=

lea)

+~

L

(

)<> 8 101 / x :!a 8xo (a)

1=1101=i

+

[1

10

L 101=1e

(x - ,a)<>

°

[~o~ (a) 0_ ~o~

Q.

X

«1 _t)a + tX)] d(l _ t)le.

X

0

Definition 5.2.4 When

is considered as a polynomial in X1o ••• , XN it is called the Taylor polynOlllial for / at a or, synonymously, the k-jet of / at a. Relllark 5.2.5 It is easy to see that if j IPI = j, then 8(3 Po ( ) 8x(3 x is the (k - j)-jet of

at a. From Taylor's theorem follows:

~

k and

P is a multi-index with

5.3. PROOF OF THE WHITNEY EXTENSION THEOREM

167

e"

Corollary 5.2.6 II I: U ~ R. is on the open set U, A is a compact subset 01 U, P,.(x) denotes the k-jet 01 I at 0, and P is a multi-index with IPI :5 k, then lim 1{fJI (x) - {fJp,. (x) 1Ix 8xfJ 8x/J

z-+,.

al lPI - k =

0

holds unilormly on A. In the early 1930's H. Whitney proved the converse of this corollary to Taylor's theorem:

TheoreJD 5.2.7 (Whitney Extension Theorem) Suppose k is a nonnegative integer, A is a closed subset oj aN, and to each 0 E A there corresponds a polyriomiol in X1o ••• ,XN 01 degree not exceeding k such that Jor each multi-index P with IPI :5 k lim 1{fJ P" (b) - atJ Po (b)\lb 8x P 8xP

"-+0

al lPI - k =

0

holds unilormly. on compact subsets oj A. Then there exists a -+ 1R with 801 g 8OtPo 8xOl (a) = 8xOl (a)

(5.9)

e le

map g :

]RN

Jor each multi-index a with

lal :5 k

(5.10)

and each a E A.

We will give the proof of the Whitney Extension Thecirem in the next section.

5.3

Proof of the Whitney Extension Theorem

The Whitney Extension Theorem 5.2.7 allows one to extend a given function on a closed set A to a e le function on all of ]RN , provided there exist prospective Taylor polynomials for the function that satisfy the a priori bounds that Taylor's Theorem requires of a C le function. The fundamental geometric tool needed for the proof of the Whitney Extension Theorem is the Whitney decomposition of aN \ A. A Whitney decomposition of an open set U is a family of closed cubes, with disjoint interiors, the union of which equals U. In addition, each cube in a Whitney decomposition of U must have its diameter bounded above and below by constant multiples of the distance from the cube to the complement of U. Whitney's original construction of what is now called a Whitney decomposition was a paragon of simplicity (see item 8 on page 67 of Whitney [1]), but not all the cubes had their diameters bounded from below by a multiple of the distance from the cube to the complement of the open set. Our version of the proof retains the simplicity of Whitney's construction while obtaining the lower bound for all cubes.

CHAPTER 5. SMOOTH MAPPINGS

168

Theorem 5.3.1 (Whitney Decomposition) Let 1:5 r 1 and 2r1 + 1:5 r2 be fixed. For any non-empty closed subset A of R,N, there exists a family W of closed cubes with sides parallel to the coordinate axes such that

= UcewC, en 0' = 0, if C, c' E W with C i: C',

(i) R,N \ A (ii) (iii)

r 1· diam(C) :5

dist(C,A) :5

r2 . diam(C),

for C E W.

Proof: Assume without loss of generality that 0 E A. Let 'Ro denote the family of 3 N - 1 cubes, aU of side length obtained by taking all possible cartesian products of N intervals chosen from the set

i,

{[-I, -1], [-1,1], [1,1]}, but omitting the N-fold cartesian product of [-1, 1] with itself. For k any integer, let 'R" denote the family

{3"C: C

E

'Ro}

of cubes. For k > 0, the cubes.in 'R" are dilations of the cubes ·in 'Ro; for k < 0, the cubes in 'R" are contractions of the cubes in 'Ro. Let 'R denote the union of all the families R".(see Figure 5.1), so 'R=

U'R".

"eZ Note that if C is a cube in Ric; then diam(C)

=

2VN3"-1,

dist(C,A)

:5

dist(C,O):5 VN3"-t,.

thus any cube C E 'R satisfies dist(C, A) :5

! diam(C) :5 r 2 • diam(C).

(5.11)

Of course, the cubes in 'R have pairwise disjoint interiors and sides parallel to the coordinate axes. Let Wo denote the family consisting of all cubes C in 'R that satisfy

r 1 • diam(C) in addition to (5.11). Set

:5 dist(C,A),

(5.12)

Bo = 'R \ Woo

Let m be a non-negative integer. Suppose the family of cubes Bm has already been formed, and suppose that for each cube C E Bm the inequalities (5.11) and (5.13) dist(C,A) < r 1 ·diam(C)

5.3. PROOF OF THE WHITNEY EXTENSION THEOREM

169

~

Figure 5.1: Initial Decomposition both hold. This is the case for 80. Let 8!,. be the set of cubes formed ·by subdividing each cube of 8 m into 2N congruent subcubes with pairwise disjoint interiors and sides parallel to the axes. If C' is one of the cubes of 8!,. formed by subdividing the cube C of 8 m , we see that diam(C')

=

~diam(C),

dist(C', A)

:5

dist(C,A)

+ diam(C')

< r 1 • diam(C) + diam(C') =

2r1 • diam(C') + diam(C')

:5

r 2 • diam(C').

So every·cube in 8!,. satisfies (5,11). Let Wm+l be the family consisting of all the cubes in 8!,. that satisfy (5.12), and let 8 m +l be the family consisting of all the cubes in 8!,. that satisfy (5.13). Thus 8 m + 1 satisfies the same condition assumed for 8 m , and the construction can continue inductively. Set 00 W=

U Wm · m=O

It is clear from the construction that every cube in W satisfies the required inequalities, but also UW exhausts RN \ A because

and liN \ A is open.

•

Definition 5.3.2 For a non-empty closed set A c aN, a collection, W, of closed cubes with sides parallel to the coordinate axes and satisfying (i)-(iji)

CHAPTER 5. SMOOTH MAPPINGS

170

0/ Theorem 5.3.1 will be called a Whitney decomposition of aN \ A or a Whitney family for A. Whitney's definition of the.extension of a function from A to aN \ A is given by a summation over all the cubes of the Whitney decomposition of JIlN \ A : For each cube the summand is a cut-off function times one of the hypothesized polynomials Pdx), where e A is a point as near as possible to the cube. The cut-off function associated with a given cube is a translation of one fixed function with support scaled to match the diameter of the cube. To see that the extension formula comes out correctly, we will need to establish some facts about that cut-off function and its derivatives. Such is our next task.

e

Definition 5.3.3 For C c JlN" a cube, let center as C, with sides parallel to the sides that o/C.

a denote the cube with the same 0/ C, but with side-length double

Lemma 5.3.4 Let 1 ~ r 1 and 2r 1 + 1 :$ r 2 be fi:xed. There e:z:ists a constant K = Krl.r2 such that whenever A is a non-empty closed subset of liN, W is a Whitney decomposition of JIlN \ A. and p e JIlN \ A, then

(i) C E W ==>

an A = 0,

(ii) card{ C E W : p E C} ~ K. Proof: (i) Arguing by contradiction, if we suppose a E anA, then we can consider the line through a and the center, c, of C and let x be the point of BC nearest to a and x' the point of BC farthest from a. Then we have

la-xl + Ix- cl = la':""cl ~ 2lx-cl = Ix-x'i <

diam(C),

hence dist(A, C)

~

la - xl < diam(C)

:$ rldiam(C),

contradicting condition (iii) of Theorem 5.3.1.

(ii) Fix p e JIlN \ A and set 6 = dist(p, A) > O. We will eatablish the existence of constants Cl and C2 such that if pea, then C

c B(p, Cl 6)

and

diam( C) 2:: C2 6.

Since the cubes in W are pairwise disjoint, we see by considering the Ndimensional volumes that it suffices to take K

= U(N) (~~)

N

NN/2,

5.3. PROOF OF THE WHITNEY EXTENSION THEOREM

171

where r N denotes the N-dimensional volume of the unit ball in JRN. Assuming pEe, we easily estimate dist(P, C} ~ ~ diam(C).

(5.14)

Also, it is clear that dist(p, A) - dist(p, C)

~

dist(C, A) ~ dist(p, A)

+

dist(p, C)

holds. By condition (iii) of Theorem 5.3.1, we also have diam(C)

~

r 1 diam(C) ~ dist(C,A) ~ r 2 diam(C).

First we obtain diam(C)

~ ~

~

dist(C, A) dist(p, A) + dist(p, C) 6 + ~ diam(C) ,

so that .diam(C)

~

26

and, by (5.14), dist(p, C)

~

6.

Clearly if r = diam(C) + dist(p, C), then C c i(p, r), so we can take Cl = 3 and obtain C C i(p, c 1 6). N ext we estimate

6

so we can take

C2

= ~

= ~

dist(p, A) dist(p,C)

~

~diam(C)

=

(~+r2)diam(C),

+ dist(C, A) + r 2 • diam(C)

•

+ r 2•

Definition 5.3.5

(i) For the remainder of this section, fix some Coo function ¢> : JR with the following properties

o ~ ¢>(t) ~ 1 for all t E IR, [-~ - E, ~

+ E] C {t: ¢>(t) =

supp¢> C (-1,1).

~

JR

(5.15) I}

for some

E

> 0, (5.16) (5.17)

:>M.UU'l'l1 M.APPlNG::;

l..I.DJU· H!df. l>.

,1/'"

(ii) For a cube C with sides parallel to the coordinate axes centered at p ='(Pl,P2, ... ,PN) and with side length s > 0, that is N

C=

II[-!+p;,!+pd, i=1

we set

-

,pc{x)

'

= lIN ,p (Xi --Pi) - . ;=1

s

Based on Definition 5.3.5, the following result 'is plainly true. Lemma' 5.3.6

(i) For any closed cube C with sides parallel to the coordinate axes and having non-empty interior, the Junction ~C is supported in the interior of C and is identically 1 on C. (ii) .If W is a Whitney family for the

non-e~pty

closed subset A, then

L:0 ~c(x) ~ 1 > 0 for all E JRN \ A, L: ~c(x) =Oforall~EA, X

(5.18) ,

CeW

(5.19)

GeW

and there exists a family of constants {I(o} (determined by the par'ticular choice of,p) such that (5.20)

holds for each multi-index Q. Lemma 5.3.7 If W is a Whitney family for the non-empty closed subset A and

then there exists a family of constants

{Ko} such that (5.21)

holds on JRN \ A.

5.3. PROOF OF THE WHITNEY EXTENSiUN 'l'Hl!JUHEM

173

For the proof of Lemma 5.3.7, we will use the formula for the derivatives . of the composition of two functions. In the one variable case, this is known as the Formula of Faa. de Bruno (see Krantz and Parks [2], Lemma 1.3.1).· For the situation at hand, we will not need the actual values of the coefficients. We leave it to the reader to use induction to verify the formula. Proposition 5.3.8 (Generalized ForlDula of Faa de Bruno) There are positive integer coefficients qA,o) such that, for each multi-index A E AN, iff and 9 are functions of class CI).1 and h = 9 0 f, then D).h(x)

=L A

II

qA,o)

g{k) [f(x)] (DO f(x»A{o) ,

oEAN

holds, where the sum extends over functions A : AN

L

-t

Z+ such that

A(a) a =A,

OlEAN

and where k = k(A) is defined by

Proof of LelDlDa 5.3.7:· Set get) = lIt and <1>(x) =

L

¢c(x),

CEW

then h = 9 0 <1>. Now, since <1> is bounded below on IR \ A by 1, we see quite simply that Ig{k) [<1>(x)] k!

I: :;

holds on IR\A. By the upper bound (5.20) and the bound on the number of non-vanishing ¢c at any point of IR \ A, we have a bound on D°<1>(x), and the lemma follows from the Generalized Formula of Faa de Bruno. • In the next definition, we construct a partition of unity for ]RN \ A. It is crucial that we can estimate the size of the partial derivatives of the functions making up this partition of unity and that all those functions vanish on A.

Definition 5.3.9 (Whitney Partition of Unity) Suppose W is a Whitney decomposition for the non-empty closed set A.

(i) For C E W, we set ¢c(x)

= ¢c(x) /

(

L C'EW

¢c' (X») for x E IRN \ A,

CHAPTER 5. SMOOTH MAPPINGS

174

so

E

q,c(x) = 1 for all x E RN

\ A,

(5.22)

CeW

E

q,c(x)

= 0 for all x EA.

(5.23)

CeW

There e:Eists a family of constants {Ko} such that (5.24)

holds for each multi-index Q.

(ii) Let

e: W

~

A be such that dist(C, A) = dist ({(C), A)

(5.25)

holds for each C E w. Of course, the choice of e(C) need not be unique; any point meeting the condition may be chosen. Remark 5.3.10 The constants Ko in (i) depend not only on the choice of 4>, but also on the value of Krt.r2 from Lemma 5.3.4. Now that we have dealt with the preHrninaries regarding the Whitney decomposition and the associated partition of unity, we can pr~ceed to the actual proof. Proof of the Whitney Extension Theorem 5.2.7: Let k be a nonnegative integer and let A be a non-empty closed set. Suppose that to each a e A there corresponds a polynomial

of degree not exceeding k, such that the limits in the hypotheses of the Whitney Extension Theorem are attained uniformly on compact subsets of A. Recall those limits are Eq. (5.9) lim 1 8PP" (b) - aPPo (b)llb- al lPH

"-'0

8xP

8xP

= o.

Choosing r 1 > 2 in Theorem 5.3.1, we let W be a Whitney decomposition for liN \A. The desired function g can simply be written down by setting

g(x) = {

P.,(x)

forxEA,

Lcew 4>c(x) p(c) (x)

for xE RN

\A.

(5.26)

5.3. PROOF OF THE WHITNEY EXTENSION THEOREM

175

The difficulty lies in showing that 9 is C le and has the polynomial P" as its Taylor polynomial of degree k at a E A. Obviously, 9 is infinitely differentiable on RN. \ A and D"'g(x)

=

L L (~)D,8cf>c(X)D"'-,8P«C)(X)

(5.27)

CEW,8S'"

for x E RN \ A and any multi-index a. For any multi-iildex a, with lal $ k, we define a function T", : RN ~ R by setting for x E A, for x E RN \A. Also, for any any multi-index a, with lal <:: k, and any x E RN, we define a linear function L",(x) : RN ~ R by setting N

(L",(x), v}

= ~)v. ei) T",.+ (x), i=1

where

for j = i, forj # i.

We will show that, if a is a multi-index with lal $ k, then T", is continuous at every point of A, and if lal < k and a E A, then lim IT",(x) - T",(a) - (L",(a), x - a}1 = O.

Ix - al

"-:>,,

This will verify that 9 is C le on all of RN and that the Taylor polynomial of 9 at any point a E A coincides with Pa. For notational convenience, in that which follows let us assume that A is compact. Then we may set

€(8) = sup {

ID"'Pa(a)-D",p,,(a)1

la _ ap"'l-k

• • 0 < la -

: a, a E A,

al $

8, 0 $

lal

$ k

}

.

By hypothesis, €(o) .j. 0 as 0 .j. O. We have broken the details of the proof up into six steps. The first step will show that T", is continuous on A. Step 1: Suppose We have

lal

$ k, and consider

a ~ a with a,a E A.

IT",(a) - T",(a)1 = ID'" Pa(a) - D'" P,,(a)1 $

ID'" Pa(a) - D'" p,,(a)1

+ IDa P,,(ii) -

DO P,,(a)l.

(5.28)

CHAPTER 5. SMOOTH MAPPINGS

176

As a ~ a, the first term in line (5.28) approaches 0 because of (5.9) while the second term approaches 0 because of the continuity of DOt Pa. The second step will show that the difference quotient goes to zero when attention is restricted to A. The estimate is almost as simple as in Step 1. Step 2: Suppose lal < k, and consider Similarly to Step 1, we have

a ~ a with a, a E A.

(5.29) (5.30) As a ~ a, the term in line (5.29) approaches 0 again because of (5.9) while the term in line (5.30) approaches 0 by Taylor's Theorem applied to DOt Pa. When we consider approaching the point a E A through points lying outside of A, the situation becomes more complicated. In Steps 3 and 4 we establish a pair of preliminary estimates that will facilitate. the rest of the argument. Step 3: Suppose lal ~ k, and consider a,a E A and x E lRN. We apply Taylor's Theorem to the polynomials DO: Pa and DOl P a at the point a to estimate

IDO: Pa(x) -

<

no: Pa(x)1

L

ID1'+O:Pa(a)_D'Y+O:Pa(a)llx-~lbl

O:5I1'I:5k-lo:l

~

€(lii - aD

'Y.

L 0:511'19-10:1

Iii _ alk-I1'I-lo:llx - ~Ibl . 'Y.

(5.31)

Comparing the summation in line (5.31) with the binomial formula, we conclude that there exists a constant r 3 such that (5.32) holds. Step 4: Suppose lal ~ k, and consider x E

]RN \

Ix - iii = dist(x, A).

A and ii E A such that

5.3. PROOF OF THE WHITNEY EXTENSION THEOREM

177

Noting that

DOt Pa(x)

=

DOt

[L

rJ>c(x) Pa(X)]

CEW

L L (~)DPrJ>C(X)DOt-PPa(X)'

=

CEW P$Ot

we see that

ITOt(x) - DOt Pa(x)1

(~)DPrJ>C(X)DOt-pP(C)(X)

L L

=

- DOtPa(x)

CEW 13$0.

~

L L (~) IDprJ>c(x) I IDo.-pp(C) (x) -DOt - i3Pa(x)l· (5.33)

CEWp$Ot

.

We will estimate the term in line (5.33). Consider

IDprJ>C(x) I IDo.-p p(C) (x) - Do.-p Pa(x)1 ' . where we may assume x E C, since otherwise DprJ>c(x) have

= O. By (5.24), we (5.34)

We now need to show that Ix - iii is comparable to diam(C) and that and I~(C) -iii can be bounded by a constant multiple of diam(C). To see this, let p E C be such that I~(C) - pi = Qjst(C, A). We have I~(C) -xl

dist(C, A) ~ Ip-dl ~ Ip-xl+lx-iil, so Ix - iii

-Ip - xl

~

dist(C, A)

~

dist(C, A) - diam(C) dist(C, A) - 2diam(C) (r 1 - 2) diam(C).

~ ~

(Recall we chose r 1

Also, we estimate

r2

•

diam(C)

~

dist(C, A)

=

Ip-~(C)I

~

Ix - ~(C)I-Ip-xl Ix - ~(C)I- 2diam(C) Ix - iii - 2 diam(C).

~

~

> 2.)

CHAPTER 5. SMOOTH MAPPINGS

178 Thus we have

(f2 + 2)diam(C) ~ 1% - iii ~ (rt - 2)diam(O)." "

(5.35)

Next, we can estimate le(O) - iii

$

le(O) - pi + l1>-zl + Iz - iii

$

dist(O,A) +diam(C) + dist(%,A)

$

f

$

(3 + 2r2 ) diam(C).

2 •

diam(C) + 2diam(C) + dist(%, C) + dist{C,A)

Thus we also have I{{C) - %1 $ I{(C) - iii + Iii - zl $ (5 + 3r2)diam(O).

(5.36)

Returning to the estimation of the term in line (5.33), we can apply (5.32) and (5.36) to conclude that

IDa-.B p~(C)(z) -

Da-.B PA(Z)

I

$ r3 E({(C),ii) [diam(C)] k-1al+I.BI.

Combined with (5.34) this shows us that there exist constants r 4 and rs such that "

In Step 5, we will complete the "demonstration of continuity of Ta.

Step 5: Suppose 101 $ k, and consider z -+ a with z E Let a E A be such that

]RN \

A.

1% - al = dist(z, A). Clearly, we have 1% - al $ 1% - ai, so Iz - iil-+ 0 and a -+ a as % -+ a. We estimate ITa(z) - Ta(a)1

=

ITa(%)-DapA(%)+DapA(z)

+ D a Pa{z) D a PA(%)I + IDa PA(z) _Da Pa(z)

$

ITa{%) -

+ IDa Pa(%) -

D a Pa{a)l.

Ta(a)1 D a Pa(%)1

(5.38)

(5.39)

5.4. APPLICATION OF THE WHITNEY THEOREM

179

As z -. a the term in line (5.38) approaches 0 by the estimate (5.37) in Step 4, the second term in line (5.38) approaches 0 by the estimate (5.32) in Step 3, and the term in line (5.39) approaches 0 by the continuity of Dapa• Step 6 will complete the demonstration of differentiablity of Ta. Step 8: Suppose lal < k. Again let ii e A be such that Iz -

iii = dist(z, A).

We estimate

ITa(z) - Ta(a) - (La (a), z - a)1 Iz-al

< ITa(z) - DOl Pc\(z)1 -

Iz-al

·IDa Pc\(z) - DOl p .. (z) I Iz-al

+~--~~--~~~

+

IDa Pa(z) - DOl Pa(a) - (La(a), Z - a)1 Iz-al··

(5.40) (5.41) (5:42)

The term in line (5.40) approaches 0 by the estimate (5.37) in Step 4, the term in line (5.41) approaches 0 by the estimate (5.32) in Step 3, and the term in line (5.42) approaches 0 by Taylor's Theorem applied to the polynomial DOl P a . •

5.4

Application of the Whitney Extension Theorem

The so-called Cantor function provides a well-known example of a function on the unit interval that is non-decreasing and non-constant, but for which the derivative vanishes almost everywhere. By a similar construction H. Whitney was able to construct a function on an arc in the plane that can be extended to a C 1 function in the whole plane with the amazing property that the entire arc consists of critical points of the function, but the function is not constant on the arc. The extension of the function from the arc to the whole plane is accomplished by the Whitney Extension Theorem 5.2.7. We base our construction on Whitney [2]. The Cantor Set We form a Cantor set C in the closed unit square [0,1] x [0,1] as follows: Fix 0 < ~ < 1/2. Let C 1 consist of four closed squares of side length ~

CHAPTER 5. SMOOTH MAPPINGS

180

-[I]jj]

-~

(a) First Stage

(b) Second Stage

Figure 5.2: Forming the Cantor Set centered in the four quart«;!x:s of the unit square, as in Figure 5.2(a). Let us denote these squares by So, Sl, S2, S3 in positive cyclical order and, to be precise, let So be the square in the bottom left quarter of the unit square. Notice that the distance from any square Si to the exterior of its quarter of the unit square is ~ (~ - ~) so, if i 'I i, then dist(Si, Sj) ~ (! - A). Let C 2 consist of sixteen closed squares of side length A2 formed by applying the preceding co~struction with each square Si taking the role of the unit square. These siXteen squares are illustrated in Figure 5.2(b). The subsquares of Si will be denoted by SiO,Sil,Si2,Si3 again labeled in cyclic order, but we will specify later whether this is to cycle in the positive or negative direction and we will specify later which subsquare is designated Sm. As before, we can estimate the distance between subsquares:. if i2 'I i~, then dist (Sit i2 , Sil i;,) ~ A (~ - ~). Continue this construction to form Cs , C4 , ••• where Cle consists of 4i squares of side length Aj, with those sqllares denoted by Sil i2 ... i.' The distance estimate extends to dist(Si,i2 ... i._, i., Sili2 ... i._l;~) ~ ~Ie-l The Cantor set is then

(! -~)

if

ile =1= i~.

(5.43)

00

C= nCIe. 10=1

Any point p E C is associated with a unique sequence (iI, i2, ...) such that 00 {p} =

n

S;I;2 ...;.·

(5.44)

1e=1

Thus we can easily define a function

i : C ~ III by setting 00

i(P) = Likpk.

(5.45)

10=1

Lemma 5.4.1 Suppose p, q E C with corresponding sequences (il, i2,"') and (il,§2, ... ). Iiil =il' i2 =§2, ... , ik=i" andile+! =l=ile+!, then

(5.46)

5.4. APPLICATION OF THE WHITNEY THEOREM and I/(P) - f(q)1

~

181

31",,.+1 •

(5.47)

-'"

•

ProoC: This follows easily from (5.43) and (5.45). Corollary 5.4.2 There exist constants 0 < a and p,q E C with P:F q, then

I/(P) - l(q)1 ~

r <

riP - ql""·

00

such that

il

(5.48)

ProoC: The corollary follows by solving for k in both (5.46) and (5.47) and comparing the results. We find log", logA'

a=--

r

•

= 1 ~'" (~ _ A)"" •

Remark 5.4.3 The estimate in Corollary 5.4.2 shows that, if we let K be the greatest integer strictly less than a, then I can be extended from C to ]R2 as "a C K function with every point of C a critical point. This is a consequence of the Whitney Extension Theorem 5.2.7 applied with the polynomial Pa at a E C chosen to be the constant polynomial 1(0). While K can be made as large as we wish by choosing IJ small, the resulting set of critical points is totally disconnected. Also, if we consider the set of numbers I(C), all of which are critical values, this is also generally a disconnected set. A crucial observation is that jf IJ then I(C) [0,1]. This choice of '" is governed by fact that we have put four subsquares in each square at every stage of the construction. ~"

=h

=

The Arc From now on we set 11.-

1

r-4

and choose

(5.49)

We describe how to run an arc A given by I{) : [0,1] -t ]R2 through the Cantor set C. The construction proceeds in stages in concert with the formation of the Cantor set. Simultaneously, we will extend I from the Cantor set C to the arc. First Stage: The following construction is illustrated in Figure 5.3(a).

(i) Starting at the point (0,0), run a line segment to the bottom left comer of the square So. Define I{) on the interval [0, II to map linearly

onto this line segment, with 0 mapping to (0,0). The value of I on this part of the arc will equal the minimum value of I on the set So n c, i.e. O.

CHAPTER 5. SMOOTH MAPPINGS

182

(a) First Stage

(b) Second Stage

Figure 5.3: Running an Arc through the Cantor Set

(ii) From the bottom right corner of 8 0 run a line segment to the bottom left comer of SI. Define I{J on the interval [i, i] to map linearly onto this line segment, with i mapping to the bottom right corner of So. The value of 1 on this segment will be the maximum of 1 on the set 8 0 n C, whicl& also equals the minimum "alue 01 1 on the set SI n C. That value is ~. (This coincidence of the maximum of 1 on So n C and minimum of Ion 8 1 n C is a result of choosing JJ = ~.) (iii) From the top left left corner of 8 1 run a line segment to the bottom left corner of S2. Define I{J on the interval [~,~] to map linearly onto this line segment, with .~. mapping to the top left corner of SI. The value of 1 on this segment is t which is the maximum of 1 on SI n C and the minimum of f on 82 n C. (iv) From the top left corner of 8 2 run a line segment to the right corner of S3. Define cp on the interval [f,~] to map linearly onto this line segment, with \ mapping to the top left corner of S2. On this segment, set 1 equal to i' the maximum of 1 on 8 2 n C and the minimum of f on S3nC. (v) From the top left corner of 8 3 run a line segment to the point (0,1). Define cp on the interval [J, 1] to map linearly onto this line segment, with mapping to the top left corner of S3. On this segment, set 1 equal to 1, the maximum of f on 8 a n C.

I

The arc cp has been defined on ~,I]u[i,i]u[t,~]u[f,~u[J,~.

Define CPl : [0,1] -+

]R2

by extending

[I,

n [I,

I{J

linearly to the missing intervals

~], [~,

f),

[~,

lJ.

Second Stage: Scale the previous construction by a factor of ~ and rotate and translate to fit it in the subsquares Si and to connect with the part of

5.5. A MULTIDIMENSIONAL FUNDAMENTAL THEOREM

183

the arc already constructed. At this time, we can determine the ordering of the labeling of the squares Sij, so the second index increases as we proceed along the arc. In the process, the definition of