hilbert space methods for partial differential equations - r. showalter

Weobtain useful linear spaces of functions on the closure G as follows:It follows that M is then and only then a linear space with addition andscalar multiplication inherited fromV.. The

Partial Di erential Equations

R E Showalter

Preface

This book is an outgrowth of a course which we have given almost riodically over the last eight years It is addressed to beginning graduatestudents of mathematics, engineering, and the physical sciences Thus, wehave attempted to present it while presupposing a minimal background: thereader is assumed to have some prior acquaintance with the concepts of \lin-ear" and \continuous" and also to believeL2 is complete An undergraduatemathematics training through Lebesgue integration is an ideal backgroundbut we dare not assume it without turning away many of our best students.The formal prerequisite consists of a good advanced calculus course and amotivation to study partial dierential equations

pe-A problem is called well-posed if for each set of data there exists exactlyone solution and this dependence of the solution on the data is continuous

To make this precise we must indicate the space from which the solution

is obtained, the space from which the data may come, and the ing notion of continuity Our goal in this book is to show that varioustypes of problems are well-posed These include boundary value problemsfor (stationary) elliptic partial dierential equations and initial-boundaryvalue problems for (time-dependent) equations of parabolic, hyperbolic, andpseudo-parabolic types Also, we consider some nonlinear elliptic boundaryvalue problems, variational or uni-lateral problems, and some methods ofnumerical approximation of solutions

correspond-We brie y describe the contents of the various chapters Chapter Ipresents all the elementary Hilbert space theory that is needed for the book.tended both as a review for some readers and as a study guide for others.Non-standard items to note here are the spaces Cm( G), V , and V0 The

G of functions on R n and thelast two consist of conjugate-linear functionals

Chapter II is an introduction to distributions and Sobolev spaces Thelatter are the Hilbert spaces in which we shall show various problems arewell-posed We use a primitive (and non-standard) notion of distributionwhich is adequate for our purposes Our distributions are conjugate-linearand have the pedagogical advantage of being independent of any discussion

of topological vector space theory

Chapter III is an exposition of the theory of linear elliptic boundaryvalue problems in variational form (The meaning of \variational form" is

explained in Chapter VII.) We present an abstract Green's theorem whichpermits the separation of the abstract problem into a partial dierentialequation on the region and a condition on the boundary This approach hasthe pedagogical advantage of making optional the discussion of regularitytheorems (We construct an operator@ which is an extension of the normalderivative on the boundary, whereas the normal derivative makes sense onlyfor appropriately regular functions.)

Chapter IV is an exposition of the generation theory of linear semigroups

of contractions and its applications to solve initial-boundary value problemsfor partial dierential equations Chapters V and VI provide the immediateextensions to cover evolution equations of second order and of implicit type

In addition to the classical heat and wave equations with standard ary conditions, the applications in these chapters include a multitude ofnon-standard problems such as equations of pseudo-parabolic, Sobolev, vis-

bound-constraints on solutions We hope this variety of applications may arousethe interests even of experts

Chapter VII begins with some re ections on Chapter III and developsinto an elementary alternative treatment of certain elliptic boundary valueproblems by the classical Dirichlet principle Then we brie y discuss certainunilateral boundary value problems, optimal control problems, and numer-ical approximation methods This chapter can be read immediately afterChapter III and it serves as a natural place to begin work on nonlinearproblems

There are a variety of ways this book can be used as a text In a yearcourse for a well-prepared class, one may complete the entire book and sup-plement it with some related topics from nonlinear functional analysis In asemester course for a class with varied backgrounds, one may cover Chap-ters I, II, III, and VII Similarly, with that same class one could cover inomit I.6, II.4, II.5, III.6, the last three sections of IV, V, and VI, and VII.4

We have included over 40 examples in the exposition and there are about

200 exercises The exercises are placed at the ends of the chapters and each

is numbered so as to indicate the section for which it is appropriate

Some suggestions for further study are arranged by chapter and precedethe Bibliography If the reader develops the interest to pursue some topic inone of these references, then this book will have served its purpose

R E Showalter

Austin, Texas

January, 1977

1 Linear Algebra 1

2 Convergence and Continuity 6

3 Completeness 10

4 Hilbert Space 14

7 Expansion in Eigenfunctions 24

II Distributions and Sobolev Spaces 31 1 Distributions 31

2 Sobolev Spaces 40

3 Trace 45

4 Sobolev's Lemma and Imbedding 48

5 Density and Compactness 51

IIIBoundary Value Problems 59 1 Introduction 59

2 Forms, Operators and Green's Formula 61

3 Abstract Boundary Value Problems 65

4 Examples 67

6 Regularity 77

7 Closed operators, adjoints and eigenfunction expansions 83

IVFirst Order Evolution Equations 95 1 Introduction 95

2 The Cauchy Problem 98

v

3 Generation of Semigroups 100

6 Analytic Semigroups 113

7 Parabolic Equations 119

V Implicit Evolution Equations 127 1 Introduction 127

2 Regular Equations 128

3 Pseudoparabolic Equations 132

4 Degenerate Equations 136

5 Examples 138

VISecond Order Evolution Equations 145 1 Introduction 145

2 Regular Equations 146

3 Sobolev Equations 154

4 Degenerate Equations 156

5 Examples 160

VIIOptimization and Approximation Topics 169 1 Dirichlet's Principle 169

2 Minimization of Convex Functions 170

3 Variational Inequalities 176

4 Optimal Control of Boundary Value Problems 180

5 Approximation of Elliptic Problems 187

6 Approximation of Evolution Equations 195

Elements of Hilbert Space

1 Linear Algebra

We begin with some notation A function F with domain dom(F) = A

and range Rg(F) a subset of B is denoted by F : A ! B That a point

x 2 A is mapped by F to a point F(x) 2 B is indicated by x 7! F(x) If

S is a subset of A then the image of S by F is F(S) = fF(x) : x 2 Sg.Thus Rg(F) = F(A) The pre-image or inverse image of a set T  B is

F;1(T) =fx2A:F(x) 2Tg A function is called injective if it is one, surjective if it is onto, and bijective if it is both injective and surjective.Then it is called, respectively, an injection, surjection, or bijection

one-to-K

of R (real number system) or C (complex numbers) The choice in mostsituations will be clear from the context or immaterial, so we usually avoidmention of it

The \strong inclusion" K  G between subsets of Euclidean space

R n means K is compact, G is open, and K  G If A and B are sets,their Cartesian product is given by AB = fa
b] : a 2 A
b 2 Bg If

A and B are subsets of K n (or any other vector space) their set sum is

such that (V
+) is an Abelian group, i.e.,

give a linear space structure on the set C(G
K) of continuous f : G! K

We normally shorten this to C(G)

(d) For eachn-tuple= (1
2
:::
n) of non-negative integers, we denote

by D the partial derivative

@j j

@x 1

1 @x 2 2

 @xn n

of order jj=1+2+ +n The setsCm(G) = ff 2 C(G) : D f 2

C(G) for all , jj  mg, m  0, and C1G = T

m 1Cm(G) are linear

D
be the identity, where

= (0
0
:::
0), so C0(G) =C(G)

(e) For f 2 C(G), the support of f is the closure in G of the set fx 2

G : f(x) 6= 0g and we denote it by supp(f) C0(G) is the subset of those

0 (G) =

Cm(G) C (G),m 1 and C1(G) =C1(G) C (G)

1 LINEAR ALGEBRA 3(f) If f :A ! B and C  A, we denote fj C the restriction of f to C Weobtain useful linear spaces of functions on the closure G as follows:

It follows that M is then (and only then) a linear space with addition andscalar multiplication inherited fromV

Examples. We have three chains of subspaces given by

boundary ofG Likewise we can identify each 2Ck( G) with j G 2CK(G)

of the coset ^x and we clearly have y 2x^ if and only if x 2y^ if and only if

The proof follows easily, sinceMis closed under addition and scalar

multipli-x+ ^y=(xd+y) and x^=(dx) These operationsmake V=M a linear space

Examples. (a) Let V = R

2 and M = f(0
x2) : x2

2 Rg Then V=M isthe set of parallel translates of thex2-axis,M, and addition of two cosets iseasily obtained by adding their (unique) representatives on thex1-axis.(b) Take V =C(G) Let x0

2 Gand M =f'2C(G) :'(x0) = 0g Writeeach '2V in the form'(x) = ('(x);'(x0)) +'(x0) This representationcan be used to show that V=M is essentially equivalent (isomorphic) to K.(c) Let V = C( G) and M = C0(G) We can describe V=M as a space of

\boundary values." To do this, begin by noting that for eachK Gthere

is a  2C0(G) with = 1 on K (Cf Section II.1.1.) Then write a given

by K(T)

Lemma If T :V !W is linear, then K(T) is a subspace of V, Rg(T) is

a subspace of W, and K(T) =fg if and only if T is an injection

Examples. (a) Let M be a subspace of V The identity iM :M !V is alinear injectionx7!x and its range is M

(b) The quotient map qM :V ! V=M, x 7! x^, is a linear surjection withkernelK(qM) =M

(c) LetGbe the open interval (a
b) inRand considerD d=dx: V !C( G),whereV is a subspace ofC1( G) IfV =C1( G), thenDis a linear surjectionwith K(D) consisting of constant functions on G If V = ' C1( G) :

V to be the linear space of all conjugate linear functionalsfrom V ! K V is called the algebraic dual of V Note that there is

a bijection f 7!f of L(V
K) onto V , where f

by f(x) = f(x) for x2V and is called the conjugate of the functional

f :V ! K Such spaces provide a useful means of constructing large linearspaces containing a given class of functions We illustrate this technique in

a simple situation

Example. Let G be open in R n and x0

2 G We shall imbed the space

C(G) in the algebraic dual ofC0(G) For eachf 2C(G Tf 2C0(G)by

Tf(') =Z

Gf '
'2C0(G):

Since f '2C0(G), the Riemann integral is adequate here An easy exerciseshows that the functionf 7!Tf :C(G)!C0(G) is a linear injection, so wemay thus identify C(G) with a subspace of C0(G) This linear injection is

C0(G

with functions inC(G) In particular, the Dirac functional x 0

x 0(') ='(x0)
'2C0(G)

cannot be obtained as Tf for anyf 2C(G) That is, Tf = x 0 implies that

f(x) = 0 for allx2G,x6=x0, and thus f = 0, a contradiction

2 Convergence and Continuity

The absolute value function on R and modulus function on C are denoted

by j j, and each gives a notion of length or distance in the correspondingspace and permits the discussion of convergence of sequences in that space

or continuity of functions on that space We shall extend these concepts to

a general linear space

2.1

A seminorm on the linear space V is a function p : V ! R for which

p(x) =jjp(x) and p(x+y)p(x) +p(y) for all2 K and x
y2V Thepair V
pis called a seminormed space

Lemma 2.1 If V
p is a seminormed space, then

(a) jp(x);p(y)j p(x;y)
x
y2V

(b) p(x)0
x2V , and

(c) the kernel K(p) is a subspace of V

(d) If T L(W
V), then p T :W is a seminorm on W

2 CONVERGENCE AND CONTINUITY 7(e) If pj is a seminorm on V and j 0, 1 j n, then P

nj =1jpj is aseminorm on V

Proof: We havep(x) =p(x;y+y)p(x;y)+p(y) sop(x);p(y)p(x;y).Similarly,p(y);p(x) p(y;x) =p(x;y), so the result follows Setting

y= 0 in (a) and noting p(0) = 0, we obtain (b) The result (c) follows

If p is a seminorm with the property that p(x) >0 for each x 6= , wecall it a norm

Examples. (a) For 1  k  n K n by pk(x) =P

(d) For each j, 0  j  k, and K  G

Ck(G) by pjK(f) = supfjD f(x)j : x 2 K, jj  jg Each such pjG is anorm on Ck( G)

is meant

Let S  V The closure of S in V
p is the set S =fx2V :xn !x in

V
p for some sequence fxn g in Sg, and S is called closed if S = S Theclosure S of S is the smallest closed set containing S:S S, S= S, and if

0 which shows that (xn+yn)!x+y Sincexn+yn 2M, alln, this impliesthatx+y2M Similarly, for2 K we havep(x;xn) =jjp(x;xn)!0,

Proof: LetT be continuous at xand " >0 Choose >

tion above and thenNsuch thatnN impliesp(xn ;x)< , wherexn !x

inV
pis given ThennN impliesq(Txn ;Tx)< ", soTxn !TxinW
q.Conversely, ifT is not continuous atx, then there is an" >0 such that foreveryn1 there is anxn 2V withp(xn ;x)<1=nand q(Txn ;Tx)".That is,xn !x inV
pbut fTxn g does not converge to Txin W
q

We record the facts that our algebraic operations and seminorm are ways continuous

al-Lemma IfV
pis a seminormed space, the functions(
x)7!x:K V !

V, (x
y)7!x+y:V V !V, and p:V ! R are all continuous

Proof: The estimatep(x;nxn) j;n jp(x) +jn jp(x;xn) impliesthe continuity of scalar multiplication Continuity of addition follows from

an estimate in the preceding Lemma, and continuity of p follows from theLemma of 2.1

Suppose p and q are seminorms on the linear space V We say p isstronger than q (or q is weaker than p) if for any sequence fxn g in V,

p(xn)!0 impliesq(xn)!0

Theorem 2.3 The following are equivalent:

(a) p is stronger than q,

2 CONVERGENCE AND CONTINUITY 9(b) the identity I :V
p!V
q is continuous, and

(c) there is a constant K 0 such that

q(x)Kp(x)
x2V :

Proof: By Theorem 2.2, (a) is equivalent to having the identityI :V
p!V
q

continuous at 0, so (b) implies (a) If (c) holds, then q(x;y)Kp(x;y),

x
y2V, so (b) is true

We claim now that (a) implies (c) If (c) is false, then for every teger n1 there is an xn 2V for which q(xn) > np(xn) Setting yn =(1=q(xn))xn, n1, we have obtained a sequence for which q(yn) = 1 and

in-p(yn)!0, thereby contradicting (a)

Theorem 2.4 Let V
p and W
q be seminormed spaces and T 2 L(V
W).The following are equivalent:

If K , then for every x 2 V : p(x)  1 wehave q(T(x))  K, hence K This holds for all such K, so  If

x2V with p(x)>0, then y (1=p(x))x p(y) = 1, so q(T(y)) That is q(T(x))  (x) whenever p(x)>0 But by Theorem 2.4(c) this

p(x) = 0, so we have  Thesestraightforward

Theorem 2.5 Let V
p and W
q be seminormed spaces For each T 2

L(V
W) we dene a real number byjTj pq supfq(T(x)) :x2V,p(x)1g.Then we have jTj pq = supfq(T(x)) : x 2 V, p(x) = 1g = inffK > 0 :

q(T(x))Kp(x) for all x2Vg andj j pq is a seminorm onL(V
W) thermore, q(T(x)) jTj pq p(x), x2V, and j j pq is a norm whenever q is

Fur-a norm

The dual of the seminormed space V
p is the linear space

V0 =ff 2V :f is continuousg with the norm

A sequencefxn gin a seminormed spaceV
pis called Cauchy if limmn !1p(xm

;xn) = 0, that is, if for every " >0 there is an integer N such that

p(xm ;xn) < " for all m
nN Every convergent sequence is Cauchy

We call V
p complete if every Cauchy sequence is convergent A completenormed linear space is a Banach space

Examples. Each of the seminormed spaces of Examples 2.1(a-d) is plete

3 COMPLETENESS 11Formnwe have p(xm ;xn)1=n, sofxm gis Cauchy Ifx2C( G), then

This shows that if fxn g converges to x then x(t) = 0 for 0  t < c and

x(t) = 1 for ct1, a contradiction Hence C( G),p is not complete

3.2

We consider the problem of extending a given function to a larger domain

Lemma Let T :D!W be given, where D is a subset of the seminormedspaceV
pandW
qis a normed linear space There is at most one continuous

T : D!W for which Tj D =T

Proof: Suppose T1 and T2 are continuous functions from D to W whichagree with T on D Let x 2 D Then there are xn 2 D with xn ! x in

V
p Continuity of T1 and T2 shows T1xn ! T1x and T2xn ! T2x But

T1xn = T2xn for all n, so T1x = T2x by the uniqueness of limits in thenormed space W
q

Theorem 3.1 Let T 2 L(D
W), where D is a subspace of the seminormedspace V
p and W
q is a Banach space Then there exists a unique T 2

L( D
W) such that Tj D =T, and jTj pq=jTj pq

Proof: Uniqueness follows from the preceding lemma Let x 2 D If

xn 2Dand xn !x inV
p, thenfxn gis Cauchy and the estimate

q(T(xm);T(xn))Kp(xm ;xn)shows fT(xn)g is Cauchy in W
q, hence, convergent to some y 2 W If

q(T(xn)) jTj pqp(xn)show T is continuous on Tpq= T pq

A completion of the seminormed spaceV
pis a complete seminormed space

W
qand a linear injectionT :V !W for which Rg(T) is dense inW andT

preserves seminorms: q(T(x)) =p(x) for allx2V By identifyingV
pwithRg(T)
q, we may visualizeV as being dense and contained in a correspond-ing space that is complete The completion of a normed space is a Banachspace and linear injection as above If two Banach spaces are completions

of a given normed space, then we can use Theorem 3.1 to construct a ear norm-preserving bijection between them, so the completion of a normedspace is essentially unique

lin-V
p Let

W be the set of all Cauchy sequences in V
p From the estimate jp(xn);

p(xm)j  p(xn ;xm) it follows that p(fxn g) = limn !1p(xn

function p :W ! R and it easily follows that p is a seminorm on W Foreach x2V, let Tx = fx
x
x
:::g, the indicated constant sequence Then

T : V
p ! W
p is a linear seminorm-preserving injection If fxn g 2 W,then for any " >0 there is an integer N such that p(xn ;xN) < "=2 for

nN, and we have p(fxn g ;T(xN))"=2< " Thus, Rg(T) is dense in

W Finally, we verify thatW
pis complete Letfxn gbe a Cauchy sequence

in W
p and for each n1 pickxn 2V with p(xn ;T(xn))< 1=n

p(xn ;x0)p(xn ;Txn) + p(Txn ;x0)<1=n+ limm

!1

p(xn ;xm)

we deduce that xn !x0 inW
p Thus, we have proved the following

Theorem 3.2 Every seminormed space has a completion

3.4

In order to obtain from a normed space a corresponding normed completion,

we shall identify those elements ofW which have the same limit by factoring

W by the kernel of p Before describing this quotient space, we considerquotients in a seminormed space

(b) If D is dense in V, then ^D=fx^:x2Dg is dense inV=M.

(c) ^p is a norm if and only if M is closed

(d) If V
p is complete, then V=M
p^is complete

Proof: We leave (a) and (b) as exercises Part (c) follows from the vation that ^p(^x) = 0 if and only if x2M

obser-To prove (d), we recall that a Cauchy sequence converges if it has aconvergent subsequence so we need only consider a sequence fx^n g in V=M

K is the kernel of p The continuity of p:W ! R implies that K is closed,

so ^p is a norm on W=K W
p is complete, so W=K, ^p is a Banach space.The quotient map q:W !W=K p(q(x)) = ^p(^x) = p(y) for all

y2q(x), soq preserves the seminorms Since Rg(T) is dense inW it followsthat the linear mapq T :V !W=K has a dense range inW=K We have

We brie y consider the vector space L(V
W)

Theorem 3.5 If V
p is a seminormed space and W
q is a Banach space,then L(V
W) is a Banach space In particular, the dualV0 of a seminormedspace is complete

Proof: Let fTn g be a Cauchy sequence in L(V
W) For each x2V, theestimate

q(Tmx;Tnx) jTm ;Tn jp(x)shows that fTnxg is Cauchy, hence convergent to a uniqueT(x)2W This

T :V !W and the continuity of addition and scalar multiplication

inW will imply that T 2L(V
W) We have

q(Tn(x)) jTn jp(x)
x2V

andfjTn jgis Cauchy, hence, bounded inR, so the continuity ofq shows that

T 2 L(V
W) withjTj K supfjTn j:n1g

To show Tn ! T in L(V
W), let " > 0 and choose N so large that

m
nN impliesjTm ;Tn j< " Hence, form
nN, we have

A scalar product on the vector spaceV is a functionVV ! K whose value

at x
y is denoted by (x
y x 7! (x
y) : V ! K islinear for every y2V, (b) (x
y) = (y
x), x
v 2V, and (c) (x
x) > 0 foreach x6= 0 From (a) and (b) it follows that for each x2V, the function

y 7! (x
y) is conjugate-linear, i.e., (x
y) = (x
y) The pair V
(
) iscalled a scalar product space

Proof: Part (a) follows from the computation

0(x+y
x+y) =((y
y); jj

2)for the scalars=;(x
y) and = (x
x) To prove (b), we use (a) to verify

applied to a pair of sequences,xn !x and yn !y inV
k  k

A Hilbert space is a scalar product space for which the correspondingnormed space is complete

Examples. (a) Let V =K N with vectors x = (x1
x2
:::
xN

Suppose V
(
) is a scalar product space and let B
k  k denote thecompletion of V
k  k For each y2V, the function x 7! (x
y) is linear,

to BV It is easy to verify that for eachx2B, the functiony7!(x
y) is

that (the extended) function (
) is a scalar product on B, we have provedthe following result

Theorem 4.2 Every scalar product space has a (unique) completion which

is a Hilbert space and whose scalar product is the extension by continuity ofthe given scalar product

Example L2(G) is the completion ofC0(G) with the scalar product givenabove

4.2

The scalar product gives us a notion of angles between vectors (In ular, recall the formula (x
y) =kxk kykcos() in Example (a) above.) Wecall the vectors x
y orthogonal if (x
y) = 0 For a given subset M of the

the set

M?=fx2V : (x
y) = 0 for all y2Mg :

Lemma M? is a closed subspace of V and M\M?=f0g

Proof: For each y 2 M, the set fx 2 V : (x
y) = 0g is a closed subspaceand so then is the intersection of all these for y2M The only vectororthogonal to itself is the zero vector, so the second statement follows

A set K in the vector space V is convex if for x
y2K and 0 1,

we havex+(1;)y 2K That is, if a pair of vectors is inK, then so also

is the line segment joining them

Theorem 4.3 A non-empty closed convex subset K of the Hilbert space H

has an element of minimal norm

Proof: Setting d inffkxk : x 2 Kg xn 2 K

for which xn d Since K is convex we have (1=2)(xn+xm) K for

We note that the element with minimal norm is unique, for ify2Kwith

kyk =d, then (1=2)(x+y) 2 K and Theorem 4.1(b) give us, respectively,

Proof: The uniqueness follows easily, since if x =m1+n1 with m1

K =fx+y :y 2Mg and use Theorem 4.3

n 2K with knk = inffkx+yk :y 2 Mg Then set m = x;n It

is clear that m 2M and we need only to verify that n2M? Let y 2M.For each  2 K, we have n;y 2 K, hence kn;yk

;2) 0, and this can hold forall  >0 only if (n
y) = 0

4.3

From Theorem 4.4 it follows that for each closed subspace M of a Hilbertspace H PM :H !M by PM : x =m+n 7! m,wherem2M and n2M? as above The linearity of PM is immediate andthe computation

soPM PM =PM The operatorPM is called the projection on M

If P 2 L(B
B P P =P, thenP is called a projection on theBanach spaceB The result of Theorem 4.4 is a guarantee of a rich supply

of projections in a Hilbert space

We recall that the (continuous) dual of a seminormed space is a Banachspace We shall show there is a natural correspondence between a Hilbertspace H and its dual H0 x2H the function fx

fx(y) = (x
y), y 2 H It is easy to checkthatfx 2H0 and kfx k H 0 =kxk Furthermore, the mapx7! fx:H!H0 islinear:

fx + z = fx+fz
x
z2H

f x = fx
2 K
x2H :

Finally, the function x 7! fx : H ! H0 is a norm preserving and linearinjection The above also holds in any scalar product space, but for Hilbertspaces this function is also surjective This follows from the next result

Theorem 4.5 Let H be a Hilbert space and f 2 H0 Then there is anelement x2H (and only one) for which

f(y) = (x
y)
y2H :

Proof: We need only verify the existence ofx2H Iff =we takex=,

so assume f 6= in H0 Then the kernel off,K =fx2H :f(x) = 0g is aclosed subspace ofH withK?

=fg Letn2K? be chosen withknk= 1.For each z 2 K? it follows that f(n)z;f(z)n 2K\K? =fg, so z is ascalar multiple of n (That is, K? is one-dimensional.) Thus, eachy2H is

of the form y =PK(y) + where (y
n) = (n
n) = But we also have

f(y) = (n), sincePK(y)2K, and thus f(y) = (f(n)n
y) for all y2H.The function x 7! fx from H to H0 will occur frequently in our laterdiscussions and it is called the Riesz map and is denoted byRH Note that

it depends on the scalar product as well as the space In particular,RH is

an isometry ofH onto H0

RH(x)(y) = (x
y)H
x
y H :

5 DUAL OPERATORS IDENTIFICATIONS 19

5 Dual Operators Identications

Theorem 5.1 If V is a linear space, W
q is a seminorm space, and T 2

L(V
W) has dense range, then T0 is injective on W0 If V
p and W
q areseminorm spaces and T 2 L(V
W), then the restriction of the dual T0 toW0belongs to L(W0
V0) and it satises

kT0 k L( W 0 V 0 )

 jTj pq :

Proof

the estimate

jT0f(x)j  kfk W 0jTj pqp(x)
f 2W0
x2V :

We give two basic examples LetV be a subspace of the seminorm space

W
q and leti:V !W be the identity Then i0(f) =f iis the restriction

of f to the subspace V i0 is injective on W0 if (and only if) V is dense in

W In such cases we may actually identify i0(W0) with W0, and we denote

W0

V Consider the quotient map q :W ! W=V where V and W
q are given

as above It is clear that if g2 (W=V) and f =q0(g), i.e., f =g q, then

f 2W andV K(f) Conversely, iff 2W andV K(f), then Theorem1.1 shows there is a g 2(W=V) for whichq0(g) =f These remarks showthat Rg(q0) =ff 2W :V K(f)g Finally, we note by Theorem 3.3 that

jqj q q ^= 1, so it follows thatg2(W
V)0 if and only ifq0(g)2W0

5.2

LetV and W be Hilbert spaces andT 2 L(V
W

T as follows: ifu 2W, then the functionalv 7!(u
Tv)W belongs to V0, soTheorem 4.5 shows that there is a uniqueT u2V such that

(T u
v)V = (u
Tv)W
u W
v V :

Theorem 5.2 If V andW are Hilbert spaces and T 2 L(V
W), then T 2

L(W
V), Rg(T)?=K(T ) and Rg(T )? =K(T) If T is an isomorphismwithT;1

2 L(W
V), thenT is an isomorphism and (T );1= (T;1)

We leave the proof as an exercise and proceed to show that dual tors are essentially equivalent to the corresponding adjoint LetV andW beHilbert spaces and denote byRV andRW the corresponding Riesz maps (Sec-tion 4.4) onto their respective dual spaces LetT 2 L(V
W) and consider itsdualT0

opera-2 L(W0
V0) and its adjointT 2 L(W
V) Foru2W andv2V wehaveRV T (u)(v) = (T u
v)V = (u
Tv)W =RW(u)(Tv) = (T0 RWu)(v).This shows thatRV T =T0 RW, so the Riesz maps permit us to study ei-ther the dual or the adjoint and deduce information on both As an example

of this we have the following

Corollary 5.3 If V and W are Hilbert spaces, and T 2 L(V
W), thenRg(T) is dense in W if and only if T0 is injective, and T is injective if andonly ifRg(T0) is dense in V0 If T is an isomorphism with T;1

2 L(W
V),then T0

2 L(W0
V0) is an isomorphism with continuous inverse

5 DUAL OPERATORS IDENTIFICATIONS 21BothRandT

with conjugate-linear functionals Moreover we have the important identity

if theRcorresponds to the standard scalar product onL2(G) For example,suppose R

(Rf)(g) =Z

Ga(x)f(x)g(x)dx
f
g2L2(G)

where a() 2 L1(G) and a(x)  c > 0, x 2 G Then, with the three

R corresponds to multiplication by the function a().Other examples will be given later

5.4

of a linear operator The theory of sesquilinear forms is analogous to that

of linear operators and we discuss it brie y

LetV K A sesquilinear form onV is aKvalued function a(
) on the product V V such that x7!a(x
y) is linearfor every y 2V and y7! a(x
y) is conjugate linear for every x 2V Thus,each sesquilinear form a(
) on V corresponds to a unique A 2 L(V
V )given by

-a(x
y) =Ax(y)
x
y2V : (5.1)Conversely, ifA 2 L(V
V

ear form on V

Theorem 5.4 Let V
p be a normed linear space and a(
) a sesquilinearform on V The following are equivalent:

(a) a(
) is continuous at (
),

(b) a(
) is continuous on V V,

(c) there is a constant K 0 such that

ja(x
y)j Kp(x)p(y)
x
y2V
(5.2)(d) A 2 L(V
V0)

Proof: It is clear that (c) and (d) are equivalent, (c) implies (b), and (b)implies (a) We shall show that (a) implies (c) The continuity of a(
) at(
) implies that there is a > 0 such that p(x)  and p(y)  imply

ja(x
y)j  1 Thus, if x 6= 0 and y 6= 0 we obtain Equation (5.2) with

K = 1= 2

When we consider real spaces (i.e.,K =R) there is no distinction betweenlinear and conjugate-linear functions Then a sesquilinear form is linear inboth variables and we call it bilinear

6 Uniform Boundedness Weak Compactness

A sequencefxn gin the Hilbert spaceH is called weakly convergent tox2H

if limn !1(xn
v)H = (x
v)H for every v 2 H The weak limit x is clearlyunique Similarly,fxn g is weakly bounded ifj(xn
v)H jis bounded for every

6 UNIFORM BOUNDEDNESS WEAK COMPACTNESS 23and j(xn 2
y)H j > 2 for y 2 s(y2
r2 s(yj
rj) 

s(yj ;1
rj ;1) with rj < 1=j and j(xn j
y)H j > j for y 2 s(yj
rj) Since

kym ;yn k < 1=n if m > n and H is complete, fyn g converges to some

y 2 H But then y 2 s(yj
rj), hence j(xn j
y)H j > j for all j  1, acontradiction

Thusfxn gis uniformly bounded on some spheres(y
r) :j(xn
y+rz)H j 

K for allz withkzk 1 Ifkzk 1, then

Proof: Let" > 0 and v 2H There is a z 2D with kv;zk < " and weobtain

j(xn ;x
v)H j  j(xn
v;z)H j+j(z
xn ;x)H j+j(x
v;z)H j

< "kxn k+j(z
xn ;x)H j+"kxk :

Hence, for allnsuciently large (depending onz), we havej(xn ;x
v)H j<

2"supfkxm k:m1g Since " >0 is arbitrary, the result follows

Theorem 6.2 Let the Hilbert space H have a countable dense subset D=

fyn g If fxn g is a bounded sequence in H, then it has a weakly convergentsubsequence

Proof: Since f(xn
y1)H g is bounded in K, there is a subsequence fx1 n g

of fxn g such that f(x1 n
y1)H g converges Similarly, for each j  2 there

is a subsequence fxjn g of fxj ;1 n g such that f(xjn
yk)H g converges in Kfor 1  k  j It follows that fxnn g is a subsequence of fxn g for which

f(xnn
yk)H g converges for everyk 1

From the preceding remarks, it suces to show that if f(xn
y)H g verges in K for every y 2 D, then fxk g

con-f(y) = limn (xn
y)H, y D , where D is the subspace of all linear

combinations of elements of D Clearly f f is continuous, since

fxn g is bounded, and has by Theorem 3.1 a unique extension f 2H0 Butthen there is by Theorem 4.5 an x 2 H such that f(y) = (x
y)H, y 2 H.The Lemma above shows that x is the weak limit offxn g

Any seminormed space which has a countable and dense subset is calledseparable Theorem 6.2 states that any bounded set in a separable Hilbertspace is relatively sequentially weakly compact This result holds in any

re exive Banach space, but all the function spaces which we shall considerare separable Hilbert spaces, so Theorem 6.2 will suce for our needs

7 Expansion in Eigenfunctions

7.1

We consider the Fourier series of a vector in the scalar product spaceHwithrespect to a given set of orthogonal vectors The sequencefvj gof vectors in

H is called orthogonal if (vi
vj)H = 0 for each pairi
j withi6=j Letfvj g

be such a sequence of non-zero vectors and let u2H For each j

the Fourier coecient ofuwith respect tovj bycj = (u
vj)H=(vj
vj)H Foreach n 1 it follows that P

nj =1cjvj is the projection of u on the subspace

Mn spanned by fv1
v2
:::
vn g This follows from Theorem 4.4 by notingthatu;

1 X

j =1

jcj j 2

=ku;un k

2+kun k

2
n1: (7.2)But kun k

 kuk

2for alln, hence (7.1) holds It follows from (7.2)that limn u un 0 if and only if equality holds in (7.1)

7 EXPANSION IN EIGENFUNCTIONS 25The inequality (7.1) is Bessel's inequality and the corresponding equality

is called Parseval's equation The seriesP

1

j =1cjvjabove is the Fourier series

of uwith respect to the orthonormal sequencefvj g

Theorem 7.2 Let fvj g be an orthonormal sequence in the scalar productspace H Then every element of H equals the sum of its Fourier series ifand only if fvj g is a basis for H, that is, its linear span is dense in H.Proof: Supposefvj gis a basis and letu2Hbe given For any" >0, there

is ann1 for which the linear spanM of the setfv1
v2
:::
vn gcontains anelement which approximatesu within" That is, inffku;wk:w2Mg< "

If un is given as in the proof of Theorem 7.1, then we have u;un 2 M?.Hence, for anyw2M we have

7.2

Let T 2 L(H) A non-zero vector v 2 H is called an eigenvector of T if

T(v) = for some 2 K The number is the eigenvalue of T sponding to v We shall show that certain operators possess a rich supply

corre-of eigenvectors These eigenvectors form an orthonormal sequence to which

we can apply the preceding Fourier series expansion techniques

An operator T 2 L(H) is called self-adjoint if (Tu
v)H = (u
Tv)H forall u
v 2H A self-adjoint T is called non-negative if (Tu
u)H 0 for all

Proof: The sesquilinear form u
v] (Tu
v)H

product axioms and this is sucient to obtain

u
v]2

u
u]v
v]
u
v H : (7.4)

(If either factor on the right side is strictly positive, this follows from theproof of Theorem 4.1 Otherwise, 0u+tv
u+tv] = 2tu
v] for allt2 R,hence, both sides of (7.4) are zero.) The desired result follows by setting

v=T(u) in (7.4)

The operators we shall consider are the compact operators If V
W areseminormed spaces, thenT 2 L(V
W) is called compact if for any boundedsequence fun g in V its imagefTun g has a subsequence which converges in

W The essential fact we need is the following

Lemma 7.4 If T 2 L(H) is self-adjoint and compact, then there exists avector v withkvk= 1 and T(v) =v, where jj=kTk

;2T2(w) Note thatkwk= andT2(w) = 2w Thus, either ( +T)w6= 0and we can choose v= ( +T)w=k( +T)wk, or ( +T)w= 0, and we canthen choosev=w=kwk Either way, the desired result follows

Theorem 7.5 Let H be a scalar product space and let T 2 L(H) be adjoint and compact Then there is an orthonormal sequence fvj g of eigen-vectors of T for which the corresponding sequence of eigenvalues f j g con-verges to zero and the eigenvectors are a basis for Rg(T)

self-Proof: By Lemma 7.4 it follows that there is a vectorv1 withkv1

Suppose the sequence f j g n

for which n = 0 Then Hn ;1

K(T), since T(vj) = 0 for j n Also weseevj Rg(T) forj < n, so Rg(T)? v
v
:::
vn ?=Hn and from

7 EXPANSION IN EIGENFUNCTIONS 27Theorem 5.2 followsK(T) = Rg(T)?

Hn ;1 ThereforeK(T) =Hn ;1 andRg(T) equals the linear span of fv1
v2
:::
vn ;1

g.Consider hereafter the case where each j is dierent from zero Weclaim that limj !1( j) = 0 Otherwise, since j j j is decreasing we wouldhave allj j j " for some " >0 But then

2

2"2for alli6=j, sofT(vj)ghas no convergent subsequence, a contradiction Weshall showfvj gis a basis for Rg(T) Letw2Rg(T) andP

bjvj the Fourierseries of w Then there is au2H withT(u) =w and we let P

cjvj be theFourier series ofu The coecients are related by

j on Hn, and since ku;

P

nj =1cjvj k  kuk by(7.2), we obtain from (7.5) the estimate

1.3 In Example (1.3.b), show V=M is isomorphic toK

1.4 Let V = C( G) and M = f' 2 C( G) : 'j @G = 0g Show V=M isisomorphic to f'j @G :'2 C( G)g, the space of \boundary values" offunctions inV

1.5 In Example (1.3.c), show ^'1 = ^'2 if and only if '1 equals '2 on aneighborhood of@G Find a space of functions isomorphic to V=M.

K(D) and Rg(D) when V = f' 2 C1( G) :

'(a) ='(b)g

1.7 Verify the last sentence in the Example of Section 1.5

1.8 Let M V for each 2A \fM :2Ag V

2.1 Prove parts (d) and (e) of Lemma 2.1

2.2 IfV1
p1 andV2
p2 are seminormed spaces, showp(x) p1(x1)+p2(x2)

is a seminorm on the productV1

V2.2.3 Let V
p be a seminormed space Show limits are unique if and only if

S 2 L(U
V))T S 2 L(U
W) and jT Sj  jTj jSj

2.8 Finish proof of Theorem 2.5

con-7 EXPANSION IN EIGENFUNCTIONS 293.4 Let V
p be a seminormed space and W
q a Banach space Let thesequenceTn 2 L(V
W) be given uniformly bounded: jTn j pq Kfor all

n1 Suppose thatDis a dense subset ofV andfTn(x)gconverges in

W for eachx2D Then showfTn(x)gconverges inW for eachx2V

and T(x) = limTn(x T 2 L(V
W) Show that completeness

of W is necessary above

3.5 Let V
p and W
q be as given above Show L(V
W) is isomorphic to

L(V=Ker(p)
W)

3.6 Prove the remark in Section 3.3 on uniqueness of a completion

4.1 Show that the norms p2 and r2 of Section 2.1 are not obtained fromscalar products

4.2 Let M be a subspace of the scalar product space V(
) Then thefollowing are equivalent: M is dense in V, M? = fg, and kfk V 0 =supfj(f
v)V j:v2Mg for everyf 2V0

4.3 Show limxn=xinV, (
) if and only if limkxn k=kxkand limf(xn) =

7.2 In Theorem 7.1, show thatfun g is a Cauchy sequence

7.3 Show that the eigenvalues of a non-negative self-adjoint operator are allnon-negative

7.4 In the situation of Theorem 7.5, show K(T) is the orthogonal ment of the linear span offv1
v2
v3
:::g.

comple-Distributions and Sobolev

Spaces

1 Distributions

1.1

We shall begin with some elementary results concerning the approximation

of functions by very smooth functions For each " >0, let'" 2C1

0 (R n) begiven with the properties

;"2);1
jxj< " ,

0
jxj ".Letf 2L1(G), whereGis open inR n, and suppose that the support off

f)G Then the distance from supp(f) to@Gis a positivenumber We extend f as zero on the complement of G and denote theextension inL1(R n) also by f " >

Proof: The second result follows from Leibnitz' rule and the representation

kfk L 2 G )

kk L 2 G )'"(y)dy=kfk L 2 G )

kk L 2 G )

by computations similar to the above, and the result follows since C0(G)

is dense in L2(G) (We shall not use the result for p 6= 1 or 2, but thecorresponding result is proved as above but using the Holder inequality inplace of Cauchy-Schwarz.)

Finally we verify the claim of convergence in Lp(G) If >0 we have a

g2 C0(G) with kf;gk L p =3 The above shows kf" ;g" k L p =3 and

we obtain

kf" ;fk L p  kf" ;g" k L p+kg" ;gk L p+kg;fk L p

 2=3 +kg" ;gk L p :

For"suciently small, the support ofg" ;gis bounded (uniformly) and

g" !g uniformly, so the last term converges to zero as"!0

The preceding results imply the following

'(x)1, x2G, and '(x) = 1 for all x in some neighborhood of K.

Proof: Let be the distance fromK to @G and 0< " < "+"0 < Let

f(x) = 1 if dist(x
K)"0 and f(x) = 0 otherwise Then f" has its supportwithinfx: dist(x
K)"+"0

g and it equals 1 onfx : dist(x
K)"0

0 (G) is the linear space of distributions onG, and we also denote

it byD (G)

Example. The space L1

loc(G) = \fL1(K) :K Gg of locally integrable

in the Example of I.1.5 That is, f 2 L1

loc(G) is assigned the distribution

Theorem 4.5 Let H be a Hilbert space and f... !y inV
k  k

A Hilbert space is a scalar product space for which the correspondingnormed space is complete

Examples. (a) Let V... and this can hold forall  >0 only if (n
y) =

4.3

From Theorem 4.4 it follows that for each closed subspace M of a Hilbertspace H PM