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
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