hilbert space methods in elliptic partial differential equations edward milton landesman

7/22/2019 Hilbert Space Methods in Elliptic Partial Differential Equations Edward Milton Landesman

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Pacific Journal of

Mathematics

HILBERT-SPACE METHODS IN ELLIPTIC PARTIAL

DIFFERENTIAL EQUATIONS

EDWARD MILTON LANDESMAN

Vol. 21, No. 1 November 1967


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PA C IFIC JOURNAL OF M AT HE M AT I C SVo. 21, No. 1, 1967

HILBERT-SPACE METHODS IN ELLIPTIC PARTIALDIFFERENTIAL EQUATIONS

EDWARD M. LANDES MAN

The purpose of this paper is to study, together withapplications, those aspects of the theory of Hubert Spacewhich are pertinent to the theory of elliptic partial differentialequations. This involves the study of an unbounded operatorA from one Hubert Space to another together with its adjointA*, its pseudoinverse or generalized reciprocal A"1, and its

- reciprocal A1 A*~. In order to carry out the results,further properties of the operators A~ and A' are developedin this paper.The concept of ellipticity of a partial differential operatoris introduced via the properties of an operator in a suitably chosenHubert Space. This Hubert Space is the one defined by theoperator Gk, that is, the operator which maps a function intoitself and its first k derivatives. It is shown that ellipticoperators are those that behave in a topological sense thesame as closed and dense restrictions of the operator Gk.Several other characterizations of elliptic operators and givenand their relation to each other is explained. This approachyields existence theorems for strong solutions of elliptic partialdifferential equations and provides methods for gaining strongsolutions from weak solutions.

The ^f~h spaces that arise from the so- called negativenorms and that have been used effectively by several authorsin the study of elliptic partial differential equations areobtained by the use of the - reciprocal of the operator Gk.Simple examples which illustrate the above theory areprovided.

1* Preliminaries* We will mainly be interested in Hlbert Spaces.We denote our H ubert Spaces by Sff, {ff, f", etc. When we write % f and Sf', they maycoincide. In ner products in each H ube rt Spacewill be denoted as follows:If we are concerned with an inner product in 3?, andxu x2 e J^7,we denote their inner product in Sf by: (xu x2)%?. For x^Sf, itsnorm will be denoted by | | B | | # . If no ambiguity is present, we willomit the subscripts


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114 EDWARD M. LANDESMAN

The orthogonal complement of & in Sf will be denoted by &L.&L is certainly a subspace of Sff.

We will use the term operator to denote a closed, dense, lineartransformation.

2* The pseudoinverse of an operator, its adjoint, and relatedresults* In this section we define the concept of the"pseudoinverse"of an operator which is of particular interest in this paper. We thenfollow with some basic results which the "pseudoinverse" shares. Inorder to define the pseudo-inverse of an operator, it is convenient tointroduce the ordered pair definition of a linear transformation. Webegin as follows:

Consider a Hilbert- Space {f' and a Hilbert- Space {f" and let^f = ^ff x 3{f" be their Cartesian product. Clearly J T i sa HubertSpace with inner product being the sum of the inner products of ifand Sf". Let se% = 3if' x {0}, ,%f = {0} x Sf".


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E L L I P T I C PARTIAL D I F F E R E N T I A L E Q U A T I O N S 115

(2 .5) &? = J ^ + J ^1 = (J*S + i f ) + ( J ^* +


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THEOREM 2.1. Let A be an operator from f to f.' Then therange &A of A is closed if and only if A*1 is bounded.

T H E O R E M 2.2. Suppose A and B are operators from >f toSuppose 2$A g 3fB and

rA ^ V~ B. Suppose further that A"

1isbounded. Then C = BA~ is bounded and \\Bx\\ ^ \ \C\\\\Ax\\ on

3* Theorems about an operator A and its *- reciprocal A'. Inthis section we are concerned with theorems which connect an operatorA and its *-reciprocal A!. Some of the theorems will be results whichrelate different norms to each other, i.e., theorems regarding theequivalence of topologies. We also characterize the domain of theoperator A! in terms of the operator A.

We begin with the following lemma, which is a "generalized"Schwarz inequality similar to that used by P. Lax in [5] and M.Schechter in [6]. This will prove useful later on.

Let A be an operator from 3f to 3ff. We have &rA Sf. Now,3?* = &A* &A*E Sf. Denote &rA 7 by


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E L L I P T I C PARTIAL D I F F E R E N T I A L E Q U A T I O N S 117for all x in the carrier ^ of A.

I n order to prove this, let y be in grA, and consider the relation\(x,y)\ = \ (Ax ,A 'y ) \ \ \Ax \ \ \ \A 'y \ \ .

I t follows that(3.4) I (x, y) I ^ k \ \ Ax || w h e r e k= \ \A 'y \ \ .Therefore y e S and(3.5) ^ g S .

Suppose, conversely, that yeS. Consider the linear functionall(x) = ( t y) for all x in 0. By (3.6), l(x) is boundedon ^A. Let x = A~ xz where z e & - i. Then from (3.6) we have(3.7) \ ( A ~ % y ) \ < * k \ \ z \ \ .Therefore, by the definition of the adjoint of A~\ we get(3.8) (A- % y) = (z, Ay) .Hence y is in ^ , and(3.9) S S ^ .Combining (3.5) and (3.9), we arrive at the theorem.

T he following corollary is a variation of a theorem which appearsin the reference [5]. The theorem, as it appears in [5], is a Differ-entiability Theorem which aids in the determination of regularity ofsolutions of elliptic partial differential equations. We will use theseresults in 7 to get strong solutions of elliptic differential equationsfrom weak solutions.

COROLLARY. Let A he an operator from


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O u r next result is a lemma that relates the norm of A' to then o r m of A.

Suppose A is an operator from ^f to ?f. We have srAan d 3rA, 0 such that(3.12) 0 < \\Bx\\ ^ M \\Ax\\for all x 0 in & . Then &rBt 3rA, and(3.13) \\A!v\\^M\\B'y\\for all y 0 in 3fB,.

Proof. F r o m (3.1) we have(3.14) I (x, y) I = I (Bx, B'y) \ \ \ Bx \ \ \ \ B'y \ \ Combining this result with (3.12), it follows that(3.15) \(x,y)\M\\Ax\\\\B'y\\where xe


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E L L I P T I C PARTIAL D I F F E R E N T I A L EQUATIONS 119h a v e &B,


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120 EDWARD M.LANDESMAN

4* Hubert Spaces associated with theoperators AandA!andrelated theorems* In this section we study theHubert Spaces fAand 3ifA, which are theHubert Spaces associated with the operatorsA andA' respectively. We then formulate some general theoremswhich relate thespaces toeach other. For simplicity weassume thatthe null space ^/f^Aof theoperator A is zero. Weproceed asfollows:Let A be an operator that takes


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ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 121

To prove this we use (4.3) and the ordinary Schwarz Inequality:= \(Ax,A'y)\ S\ \ Ax\ \ \ \ y \ \

i i \ \Aiiy lU'T h e following theorem is a Representation Theorem and can be

used in order to obtain existence theory in Elliptic Partial DifferentialE q u a t i o n s .

T H E O R E M 4.2. (Representation Theorem.) Every bounded linearfunctional l(y) over 3(fA, can be represented as(4.5) l(y) = where x e SfA.

Proof. Since 3(fA, is a Hubert Space, we have by the Riesz-Representation Theorem that(4.6) l(y) = (z, y)*A,where z e ^ ^ , . By the way we have constructed lfA, we can choosex e ?A such that Ax = 'z. Then

(z, y)*rA, = (A'z, A!y) = {Ax, A'y) = .Therefore l(y) x, yy where x e fA and th e theorem is complete.

5* An example* In this section we present an example thatwill serve to illustrate some of the definitions and theorems frompreceding sections. Part of the example appears in the reference[3] .

Let ^^ be the class of all real valued Lebesgue square integrablefunctions x(tu t2) on the square 0 ^ t1 S >0


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Let A be the gradient operator written Ax = grad x whose domain3? consists of those x in s$f such that x(0, t2) = x( , t2) = 0 foralmost all t2 on 0 ^ t2 ^ and ^, 0) = #(j., ) = 0 for almost all tton 0 ^ i ^ . Then A is a closed, dense operator from Sf to < ^'where (?' is the Hubert Space defined by the Cartesian productf x Sf. The adjoint A* of A is given by

A*y = divergence ywhere divergence y is the closure of the divergence operator definedon all those y in f' of class C". Since the ranges of A and A* areclosed, A and A* are reciprocally bounded, i.e., A"1 and A' are bounded.Observe that the operator

I n order to derive integral representations for A"1 and A', considert h e functions

o(5.2) (*i,


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E L L I P T I C PARTIAL D I F F E R E N T I A L E Q U A T I O N S 123a? b

.X = 2~ ^mn^mnVmn j A y J li ^mraCO ~ CO4 ' . V __ t2L oy y 4 * l / V \ for

A*AX = ^ A , Xm , n = l m yn= l

(A*A)- 1x=

Consider the following functionG(su s2, lf 2)

/ g rj\ 4 1 .v # ; = sn s n m s s i n nsz s n m^i s i R ^ ^ 9 2 m,n=i ( m2 + n2)which converges uniformly and absolutely. This is the Green's functionfor the negative of the Laplacian. When plausible, we use the condensedn o t a t i o n s = (sl9 s2), t (tu Q and (s, ) == (su s2, i, 2). We observet h a t

(5.8) G(s, t) - G(, s); G |(s, t) - G t |( , s)for i 1,2, whenever the derivatives exist. The derivatives aregiven by the formulas

Gs (s, ) = ~ cosmsj. sin ns sin m t sin nL1 Tj2 ,n= i ( m2 + n2)

(5.9)Gs (s, t) =

si

R m si

c o s nsz

si

n m^i

si

n n^2 T 2 w, =i (m2 + n2)

where the sums are taken in the mean of order two on the interval0 ^ s ^ , 0 ^ t i ^ ; i = l,2. Define

K1(8,t) = G.(8,t) = Gtl(t,s){ } K2(s, t) = GS2(s, t) = Gti(t, s)ly = (Vu1/2) e Sf\ t h e n by (5.2), (5.6), and (5.10),(5.11) A- y(t) - [K^S, t)yi(s)ds + i 2(s, t)y2(s)ds

J o Jofor almost all t on 0 g t < . Similarly, if y(t) = (j/^), 2/2(*))', t h e n


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= I Ktf, s)x(s)ds ,(5.12) 1

y t( t) = K2(t, s)x(s)dsJ ofor almost all t on 0 g t ^ . Define(5.13) H ^ ( 8 91)= i , *where i, i = 1, 2. From (5.4) and (5.13) we have

4iJ n(s ) f m,n= i + n2)2H12(s, t) =

r2 m,=i (m2(5.14)

cos ms1 sin fis2 cos mj. sin nUH12(s, t) = ^ ^ sin ms1 cos ns2 cos mt1 sin r2 i (m2 + w2)2if21(s, ) = - - ^ V cos mst sin ^s2 sin mtt cos 27 2 ,= i (m2 + n 2 ) a

4 ^ 2i J 2 2 ( s t) = _ 2 sin mSi cos ns2 sm mx cos nt2 .TZ:2 ,n=i (m2 + n2)2T h e ii i j( s, ) are the Green's functions for the operator AA* given

by (5.6). We have for y(t) = (^(), %()) e- [H12(s, t)y,(s)ds

(5.i5) : :- H*\s, t)J o

We observe that the kernel functions K^s, t) and K2(s9 t) of theoperators A~1 and A' are related to the functions Hj(s, t) as follows:GH{s, t) = Kfa t) - -H l { (8 , t) - H(8, t)

( * } Gu(8, t) = ^ ( , 8) - - H*{8, t) - H%{8, t)for i = 1, 2.

T h e one-dimensional analogue of the above example with A beingt h e differential operator d/dt follows similarly.

6. The operators Gk, Gk and negative norms. In order todefine ellipticity in terms of certain operators in Hubert Space it isconvenient at this time to define a new operator which we denote byGk. This operator has been introduced by M. R. Hestenes in his paper[ 3 ] . We proceed as follows:

Let be a bounded, connected, open set in Euclidean m-spacewith boundary d . Let a = (au a2, * , :M) where the at are non-negative integers and | a \ = a^ + a2 + + a*. Define ^fkn to be


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E L L I P T I C PARTIAL D IF FE REN TIAL EQUATIONS 125t h e class of all Lebesgue square int egrable complex- valued functionsy i ( t ) w h e r e t = (tu - - , t m ) e ; j = l, - - , n ; \ a \ k . T h e c l a s s fk *with

(6.1) (y,z)s,* = ^y(t)z%t)dt ,where j ^ n, \ a \ ^ k as its inner product forms a Hubert Space overt h e field of complex numbers. H enceforth, denote (y, z)^ by (y, z) ko r , when clear, by {y, z). Furthermore, by yk(t) we will mean ally a ( t ) such that |a \ = k.Define theoperator

= ^where again a = (aly , am), cti are nonnegative integers and | a: =aL + + w . For |a \ = 0, define jD = I , the identity operator.We nowdefine a closed subspace ^ of J?fkn as follows: Let &be the set of all x(t) of the form x ja( t) in ^ J % such that if we setx j( t ) = xi(t), then xja{t) = Daxj(t).T h e subspace ^ f then consists of those functions and all theirderivatives up to order k which are in J2^\ Hence, a mapping is definedfrom ^fQn to fkn which maps a function xo(t) into the set consistingof xo(t) and all of it s derivatives of order less than or equal to k.We call this mapping Gk. Therange & k of Gk is &f% and the domain3t&k of Gk is the projection of 2$% on ^ % . By the way we havedefined Gk, it is clear that Gk is a closed and dense linear transforma-t ion from ^fon to ^fkn. Theoperator G fc is one- to- one, i.e., the nullspace of Gk is zero. We note that G o = J, the identity mapping.I n recent years several authors have introduced the "negativen o r m s " in order to deal with problems in Elliptic Partial- D ifferentialE q u a t i o n s . See, for example, the references [5] and [6],

P . Lax and M. Schechter have used the following definition fort h e "negative norms." Forx e C,(6.2) \ \ x \ \ _ k i ^ ^1 2 / 1 1 *where the supremum is taken over all y e C. H ere (x, y) denotes theo r d i n a r y ^f- inn er product and \ \y\\ k the Dirichlet norm(6.3) 111/111= ( \Va(t)\ 2dt.Observe tha t(6.4) \ \ G k y \ \ = \ \ y \ \ k .


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126 EDWARD M. LANDESMANThe in teresting relation between the "negative n orms" and th e operatorGk is expressed in the following

T H E O R E M 6.1. ||G[x | = ||x ||_fc.The proof follows immediately from Lemma 3.2.We denote theHubert Spaces which are the completions of the

C-functions under the Gk norm and Gk norm by fk and 3fL krespectively.Using therelations that we have derived between 3fA and 3(fA, in 4 for an arbitrary operator A, we observe t ha t for A= Gk andA' = G'k, therange of theoperator Gk, which is thesubspace 2$k offkn, is in one- to-one correspondence with theelements of ^fk aswell

as with theelements of 3f_k. Similarly, theclosure of therange ofGfc, which is also & % , is inone-to-one correspondence with the elementsof ^fk as well aswith the elements of^f_k. The norm of the elementsin gffh and in f_k is taken to be theordinary Dirichlet norm.Finally, we observe t ha t the elements in 3fk are in one-to-onecorrespondence with theelements in

7* An o t h er definition of ellipticity* We begin bydefining theconcept of ellipticity as is done most often in thel i terature. We dothis for systems of equations.

As before, let be a bounded, connected, open set in Euclideanm- space anddenote its boundary byd . For the present wewill maken o assumptions on theboundary. Let t (tl9 ,*) denote a pointin . Suppose a = (alf ,am) where thea{ are nonnegative integersand denote by |a | the sum | | = + + am. Define

(7.1) Dax = . * ' * ' * ^ .H e r e andelsewhere, unless otherwise indicated, repeated indices willbe summed on. Consider the system(7.2) Ax =p

j =1, . . - , w = 1, - ,? .

By theprincipal part of Ax will be meant(7.3) Px =pls(t)Dax*{t) \a\ = k.A is said to beelliptic if at each point t e ,(7.4) p J{t) aV = 0 I I = fc


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E L L I P T I C PARTIAL D I F F E R E N T I A L E Q U A T I O N S 127holds for nonnull complex numbers = { \ , n) and real numbers = (1, A, , .) only in case = (0, 0, . . , 0) where

An example of the above for a system of m elliptic differentialoperators with one function x(t) is the following(7.5) Ax = Gradient x = ($L fL , ...,J?L

Writing (7.5) out in terms of a system, we have ~ d% 2~ dx - dx(7.6) AH

Here(7.7) Px = A*x i = 1, , m .Now, the characteristic polynomials are lf , and the condition(7.4) implies that & = 2 = . . . = m = 0, i.e., = (0, , 0). Weobserve here that when one function x(t) is involved, it is unnecessaryto introduce the = ( \ , ).We now wish to define ellipticity in a different way than it wasdefined above. We then proceed to explain the relationship betweenthis definition and the other. The following definition is advantageousin the sense that it deals with the properties of operators in HubertSpace. Cf. [3], page 1355. We begin as follows:

Let C be a bounded operator from 2$% (k > 0) to a Hubert SpapeSuppose that Bk is a closed and dense "restriction" of Gk in

= .2?*. The product Ak == CBk defines a dense linear transforma-tion. We will say that Ak is an elliptic operator of order k if Ak isclosed.An example of a "restriction" of Gk is the following: Suppose Bkis defined to be Gk operating on the closure of the subclass of jfkn,whose elements are continuous and have x(t) = 0, | a | < k on the

boundary of . We will see at the end of this section how thisparticular "restriction" is in fact one which is essential in order torelate this definition of ellipticity to the classical one.Another example of Bk is the "restriction" of Gk to all thosefunctions which can be extended to be periodic functions on a givenperiod- parallelogram. We will have occasion to use this in Theorem7.3.

The next theorem characterizes elliptic operators and can be usedas an equivalent definition for them. Cf. [3], page 1356.


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T H E O R E M 7.1. If Ak is a differential operator of order k suchthat(7.8) Bkx || ^ lAkx \ \ x | 0. Then there exists a bounded operator C suchthat Ak CBk and Ak is closed. Conversely, if Ak = CBk is closed,then (7.8) is valid for every x in the domain 3B]i of Bk.

Proof. By (7.8), &Bh S &Ah and %yTBk S ^ k . The operator Bkxis bounded, since Gk is. By Theorem 2.2, the operator AkBkx isbounded. Set C = AkBk\ Then CBk = AkB^Bk = Ak. We now show

Akxn-

and \ \Ak(xn- xm)\ \ -+O

0as m, ti > oo. Therefore J 3 ^ converges to some element, say z, andj fc being closed, implies that(7.12)And so, Akx = C5&x = C, i.e.,Cz and from (7.9) we have(7.13)

t h a t Ak =

(7.9)T h e n(7.10)as m,n>( 7 . H )

CBk is closed.

lla . - a . loo. By (7.8),

e 3A]c. Furthermore, Akxn CBkxn

= Cz.Therefore, Ak is closed.Conversely, suppose Ak = CBk is closed. Consider the mapping oft h e pairs {(x, Bkx)} in the product- space(7.14) 1 0 C: {(x, Bkx)} - {(a?, Akx)} .T h e operator 1 0 C is a bounded, one- to-one mapping of the subspace{(x, Bkx)} onto the subspace {(x, Akx)}. Since A^ and Bk are closed,{(a?, AA )} and {(x, Bkx)} are complete metric spaces. By the Open-Mapping Theorem, the inverse of / 0 C exists and is continuous, i.e.,t h e r e exists a constant hx > 0 such that(7.15)Therefore


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E L L I P T I C PARTIAL D I F F E R E N T I A L E Q U A T I O N S 129(7.16) II Bkx || || a? || + II Bkx \ \ x[|| x || + || Akx ||] .F u r t h e r m o r e ,(7.17) hJi\ \ x\\ + \ \ A k x ||] = ^ [ | | x \ \ + \ \ CB kx \ \ ] ^ h 2 1 | Bkx\ \where h2 = h,(l + | |C | |) .

Combining (7.16) and (7.17), we obtain (7.8).An immediate consequence of Theorem 7.1 is the following:COROLLARY. If the null space of the elliptic operator Ak is zero,then Ak and Bk generate the same topology.Proof. Since the null space of A

kis zero, the first half of the

inequality (7.8) is equivalent to the statement(7.18) H B ^ I I ^ M A ^ I Ifor some constant hz greater than zero. Furthermore, since Ak = CBkwhere C is bounded, we have(7.19) H A ^ I I - \ \ C B k x\ \ K \ \ B kx\ \where h4 || C|| is greater than zero. Combining (7.18) and (7.19), weobtain(7.20) )|Bkx || h31| Akx || ^ h 3h 41| Bkx || .

The above corollary expresses the following important statement,namely, that if Ak is an elliptic operator of order k whose null spaceis zero, then ||Afc&|| and ||jBfcc|| are equivalent norms where Bk is aclosed and dense restriction of Gk, and so the knowledge of Gk relaysmuch about the nature of Ak.The following theorem illustrates the relationship between theclassical definition of ellipticity and the above definition. I t is to befound in the references [3] and [4]. For a proof of this theorem,see [4] by M. R. Hestenes.

Consider the system(7.21) A%x(t) = P'Blx(t) = pJ(t)Da i{t)where = 1, , q; j = 1, ,n; \ a \ ^ k. Suppose further that thecoefficients p5j(t) are continuous functions of t on the closure of them-dimensional region . Then

THEOREM 7.2. Given a point te , suppose there is no nonnullset of real numbers = ( u , w) and no nonnull set of complexnumbers = ( \ , rf) such that the relations


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130 EDWARD M. LANDESMAN5 - o

where = 1, , q; \ a | = k holds, and a = ?1 . Suppose fur-ther that Bk is the restriction of Gk defined on the closure of thesubclass of Sfkn which consists of those elements which are continuousand have Dax\t) = 0, |a | < k on the boundary of . Then Ak iselliptic (in our sense) and conversely.

Theorem 7.2 states that ellipticity in the classical sense is equiv-alent to ellipticity in this "new" sense if Bk is restricted to thosefunctions whose derivatives up to order k vanish on the boundary of .

An application of the definition of ellipticity is that of gainingstrong solutions from "weak solutions" in elliptic partial differentialequations. First we need

D E F I N I T I O N 7.1. If A is an elliptic operator, then x is said to bea weak solution of Ax = y if(7.22) [x, A*u) = [y, u)for every u in the domain &A* of A*.Normally, the set {u} is not the whole domain of A* but is suchthat the closure of the restriction of A* to {u} is A* itself. In appli-cations, there usually exists a set of functions of class Cof this type.

As an application of some of the preceding results we prove theexistence of strong solutions from weak solutions. We assume theunderlying domain is a period-parallelogram.

THEOREM 7.3. L et Ak be an elliptic operator of order k whosenull space is zero. If x is a weak solution of Akx y, then x is infact a strong solution of Akx y.Proof. Since Ak is elliptic, Ak = CBk where C is bounded and Bkis the restriction of Gk to those functions which can be extended tobe periodic functions defined on the period- parallelogram. Clearly,

& k = &Bk and &r ,k - 3fB,h.Let v = Atu. By (7.22),(7.23) I (x, A t u ) \ = \ ( , v ) \ = \ ( y , u ) \ = \ (y, A r k v ) \ .By the corollary to Theorem7.1,(7.24) \ \ B k z \ \ S m \ \ A k z \ \ S M \ \ B k z \ \for constants m,M > 0. By Theorem 3.3, we have


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ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 131(7.25) \ \Akv\\m\\B'kv\\M\\Akv\\ .Using the Schwarz inequality, (7.23), and (7.25), we have:(7.26) \ ( x , v ) \ || y || || A[v \ \ m\ \ y \ \ \ \ B'kv \ \for all v of the form v = Au, that is, for all v e &A>k = &B>k. ByTheorem 3.1, % 2&Bk, that is, x is a strong solution of Akx = y.As a more substantial application using the operators Gk and G'k,one can get existence and regularity theory for certain classes ofpart ial differential equations by using the methods of P . Lax [6],For a treatment of this, see [5].

REFERENCES1. A. Ben- Israel and A. Charnes , Contributionsto the theory of generalized inverses,J. Soc. Indus t . Appl. Ma t h . 11 (1963), 667-699.2. M. R. Hestenes, Applications of the theory of quadratic forms in Hilbert- Space tothe calculusof variations, Pacific J. Ma t h . 1 (1951), 525-581.3. f Relative self- adjoint operatorsin Hilbert- Space, Pacific J. Ma t h . 11 (1961),4 1 Quadratic variational theory and linear elliptic partial differentialequations, Trans. Amer. Ma t h . Soc. 1O1 (1961), 306-350.5. E. M. Landesman, Hilbert- Space Methods in Elliptic Partial Differential Equations,P h . D. Dissertation, University of California, LosAngeles, 1965.6. P. Lax, On Cauchy's problem for hyperbolic equations and the differentiability ofsolutions of elliptic equations,Comm. Pure Appl. Ma t h . 8 (1955), 615-633.7. M. Schechter, Negative norms and boundary problems,Ann. of M a t h . 72 (1960),581-593.8. A. E. Taylor, Introduction to Functional Analysis, John Wiley and Sons, Inc.,New York, 1963.

Received October 15, 1965. This paper representsa part of theauthor's doctoralthesis. The a u tho r wishes to record his indebtednessto Professor M. R. Hestenes,under whom the work was done. The paper was sponsored in part by theOfficeofNaval Research, the U. S. Army Research Office (Durham), and the Nat iona l ScienceFounda t ion . Reproduction in whole or in part is permitted for anypurpose of theUni ted States Government.UNIVERSITY OF CALIFORNIA, LOS ANG E L E SAN DUNIVERSITY OF CALIFORNIA, SANTA CRUZ


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PACIFIC JOURNAL OF MATHEMATICSEDITORS

H. SAMELSONStanford University-Stanford, CaliforniaJ. P. JANS

University of WashingtonSeattle, Washington 98105

J. DUGUNDJIUniversity of Southern CaliforniaLos Angeles. California 90007RICHARD A R E N SUniversity of CaliforniaLos Angeles, California 90024

E. F. BECKENBACHASSOCIATE EDITORS

B. H. NEUMANN F WOLF K. YOSIDA

SUPPORTING INSTITUTIONSUNIVERSITY OF BRITISH COLUMBIACALIFORNIA INSTITUTE OF TECHNOLOGYUNIVERSITY OF CALIFORNIAMONTANA STATE UNIVERSITYUNIVERSITY OF NEVADANEW MEXICO STATE UNIVERSITYOREGON STATE UNIVERSITYUNIVERSITY OF OREGONOSAKA UNIVERSITYUNIVERSITY OF SOUTHERN CALIFORNIA

STANFORD UNIVERSITYUNIVERSITY OF TOKYOUNIVERSITY OF UTAHWASHINGTON STATE UNIVERSITYUNIVERSITY OF WASHINGTON* * *AMERICAN MATHEMATICAL SOCIETYCHEVRON RESEARCH CORPORATIONTRW SYSTEMSNAVAL ORDNANCE TEST STATION

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Pacific Journal of MathematicsVol. 21, No. 1 November, 1967

Friedrich-Wilhelm Bauer, Der Hurewicz-Satz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

D. W. Dubois, A note on David Harrisons theory of preprimes . . . . . . . . . . . . 15

Bert E. Fristedt, Sample function behavior of increasing processes with

stationary, independent increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Minoru Hasegawa, On the convergence of resolvents of operators . . . . . . . . . . 35

Sren Glud Johansen, The descriptive approach to the derivative of a set

function with respect to a -lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

John Frank Charles Kingman, Completely random measures . . . . . . . . . . . . . . 59

Tilla Weinstein, Surfaces harmonically immersed in E3 . . . . . . . . . . . . . . . . . . . 79

Hikosaburo Komatsu, Fractional powers of operators. II. Interpolation

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Edward Milton Landesman, Hilbert-space methods in elliptic partialdifferential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

O. Carruth McGehee, Certain isomorphisms between quotients of a group

algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

DeWayne Stanley Nymann, Dedekind groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Sidney Charles Port, Hitting times for transient stable processes . . . . . . . . . . . 161

Ralph Tyrrell Rockafellar, Duality and stability in extremum problems

involving convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Philip C. Tonne, Power-series and Hausdorff matrices . . . . . . . . . . . . . . . . . . . . 189
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hilbert space methods in elliptic partial differential equations edward milton landesman

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