Nonlinear Operators and the Calculus of Variations Summer School Hel

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Vol. 344: A. S. Troelstra (Ed1tor), Metamathematicallnvestigation of lntuit1onistic Arithmet1c and Analysis. XVII, 485 pages. 1973.

Vol. 371: V. Poenaru, Analyse Diff o . Then

and of the dual operator

r(T)

is an

T ~ with eigenvector8 in

P

P~ , respectively.

I t is well-known that much more precise results can be obtained i f the class of positive endomorphisms is further restricted. In applications to problems in analysis i t turns out that an important and useful subclass is given by the class of strongly positive and almost strongly positive linear operators. Let

(E,P)

and

(F,Q) be OBSs such that

T : E -* F is called strongly positive i f positive (a.s.p.) i f

o

Q* ~ . A linear operator o T(l5) c Q and almost strongly o

P\ ker T # # and T(P\ker T) c Q .

10 O

Suppose that

P ~ ~ and l e t

eigenvector of

T ~ L(E)

be a.s.p.. Then every positive

T tO a positive eigenvalue belongs to

~ . The following

lemma contains a related, somewhat weaker property for positive eigenvectors of

T~ .

(1.12) Len~na= Suppose that exist is,

p > 0

and

is a.s.p, and suppose that there

T~@ = pC . Then @(P',ker T) > o

@E p~( w i t h

(that

@(P\ker T) c ~+ ). O

Proof: Let

o . Hence

r(T) , and no solution i n If

X = r(T)

and

P\ker T i f

x ~ P if

~

o .

an a r b i t r a r y p o s i t i v e eigenvector o f

(cf. Theorem ( 1 . 1 1 ) ) . Then

r ( P \ k e r T) > o

by Lemma (1.12). (ii)

Suppose t h a t

xE P\ker T

and

x o , then there e x i s t s an element y E p

r(S)y = Sy ~ Ty . Hence r ( S ) < r Consequently,

= < l ~ , y >

such t h a t

= r(T)

r(S) ~ r(T) . The remaining p a r t o f the assertion f o l l o w s by

a s i m i l a r consideration

9

12 I t should be noted that under the hypotheses of Theorem (1.13) i t can be shown that

r(T)

is a simple eigenvalue and that

only eigenvalue of the complexification of radius

r(T)

r(T)

is the

T lying on the c i r c l e with

(cf.[7,18,19]).

In the remainder of this paragraph we indicate some

Applications to elliptic boundary value problems: Let ~ be a nonempty bounded domain in C%)manifoldsuch that words, ~

~N

whose boundary,

r , is a smooth (that i s ,

~ lies l o c a l l y on one side of

r .

(In other

is a compact connected N-dimensional differentiable manifold

with boundary r .) We denote by A a d i f f e r e n t i a l operator of the form N

Au := -

N

z aikDiDku + .Diu + aoU i,k=l i~1 al '

with smooth coefficients and a uniformly positive d e f i n i t matrix

coefficient

(aik) . (For much weaker regularity hypotheses c f . [ 6 , 7 ] ).

(1.14) Example (T~ Dirichlet Problem): We consider the linear BVP Au = f

(2)

8oU

where BoU := u l r g E C2+U(s

= g

in

R ,

on

r

,

denotes the D i r i c h l e t boundary operator and f E C~(~) ,

for some u E (o,1) . By a solution we mean a classical

solution. Suppose that unique solution

ao ~ o . Then i t is well-known that the BVP (2) has a u = So(f,g ) E C2+U(~) . Moreover, the maximum principle

and the Schauder a p r i o r i estimates imply that the operator to

L+(CU~) x C2+P