Math. Z. 179, 237 239 (1982) Mathematische Zeitschrift

�9 Springer-Verlag 1982

On the Principal Eigenvalue of a Second Order Linear Elliptic Problem with an Indefinite Weight Function

Peter Hess

Mathematics Institute, University of Ziirich, Freiestral3e 36, CH-8032 Ztirich, Switzerland

In troduct ion

We consider the Dirichlet problem

5~u=2mu in ~?, u=0 on c?f2, (l)

in the bounded domain ~2clR N (N>I ) having smooth boundary ~?f2. Here 5~ is a uniformly elliptic differential expression of second order,

N 82u N 8u

Y u = - ],k=lE ask kCOXj ~?x-4- S=I2 as =~+aoUiJxj

having real-valued coefficient functions ask=akS, aj, a o>0 belonging to C~ 0 < 0 < 1 ; further msC((2) is a given real-valued weight function, and ,le~2 the eigenvalue parameter.

By the maximum principle, (1) admits a positive eigenvalue having a positive eigenfunction only provided m is positive somewhere in ~2. We have shown in [1] that this condition is also sufficient: if re(x)>0 at some xe~2, then (1) admits a principal eigenvalue 21(m)>0, characterized by being the only positive eigenvalue having a positive eigenfunction. Moreover 21(m) is algebraically simple (in a suitable sense), and if 2~C is an eigenvalue of (1) with Re2>0, then Re2>2~(m). The last statement can be sharpened as fol- lows.

T h e o r e m . 21(m ) is the only eigenvalue 2 of (1) with Re2=Az(m ).

This Theorem has been proved by Gossez-Lami Dozo [2] under additional regularity hypotheses on m and the coefficients of Y. It seems that their method necessitates such assumptions, and that approximation techniques fail to prove the assertion in our generality. Our short and simple proof is based on results in [1].

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238 P. Hess

Proof of the Theorem. We introduce the real Banach spaces E~=C0((2 ) :={veC(~) : v = 0 on 3~1} and X : = CX(~),={veCl(~): v = 0 on 0~2}, naturally ordered by the positive cones PE and Px consisting of the (pointwise) non- negative functions. Let L denote the realization of 5~, subject to zero Dirichlet boundary conditions, in the space C(~). Then L is a closed operator with domain D(L) c X . Let further M be the multiplication operator by the function

A

m. By L, M we denote the operators obtained from L, M by complexification. We define the function u to be a solution of (1) provided

L u = 2~lu. (2)

By rescaling we may assume that ]m[ < 1 on O. For )~>0, (2) is equivalent to the equation

u = ) ~ ( L + 2 ) - l ( m + l ) u

in the space E. Let K#---(L + 2)-1(M + 1); it is an irreducible compact positive operator in E, mapping Pc'-.{0} into the interior of Px- Let u l > 0 denote the principal eigenfunction to the eigenvalue 21 , ='~l (m). Of course

ul =)~1 K~, u 1. (3)

Suppose now 2 e C is an eigenvalue of (2) with Re2=2~ , and u(=t=0) as- sociated eigenfunction. By [1, Lemma 3] we have

lu] <(Re 2) KRe ,~ ]U[.

Set v'.--X,K~.~lu], and note that veInt(Px) , lul<v. Hence v<)~lKi~v by the positivity of K;~. Let the number ~r>0 be such that w , - - v - a u l s 3 P x. Recall- ing (3), we get

w < 2 i K~., w. (4)

Two cases might occur.

(A) w>0. Let K~: E*-*E* be the Banach space adjoint operator, and u * > 0 the eigenfunction of K~ to the eigenvalue ,~i~1, guaranteed by the Krein- Rutman theorem. We have z , - - - ~ l K ~ w - w > O by (4), since )~lK~weInt(Px) and we?P x. Hence

= <)'~l K ' u * , z) = <u*, 21K;, z ) > 0

(e.g. [1, p. 1009]). This contradiction shows that case (A) is impossible.

(B) w=0. Then 2 i K~(~ui ) = au i = v =)~1K~ lu[,

and since Kz~ is injective, tu[ = Cu~. The assertion is now a consequence of (3) and the following

Lemma. Suppose u is eigenfunction of (1) to the eigenvalue )~G with R e 2 > 0 , and suppose

lu} = (Re 2) gRe ). tUl" (5)

Then 2=21, and uespan[ul~ ].

The Principal Eigenvalue of a Second Order Linear Elliptic Problem 239

Proof of the Lemma. (5) implies that (Re2)>0 is eigenvalue of problem (1) with positive eigenfunction luI. Thus, by uniqueness, Re)o=21 and (up to a scalar factor) tul=ut. We write

U~H 1 I)~

where the function v is defined in f2 and Ivl--1. Since u,u~W2'p(f2) (p>N) and u 1 >0 in O, we conclude that v=u/ulEVv~zo;v(f2). Let s denote the differ- ential expression obtained from 5f by omitting the term of order zero. Then

N 3u 1 Ov 2mulv=Sq(ulv)=UxY" v+vY 'u l -2 j , k=x ~ ajk ~?xj ~x k" (6)

We further introduce the differential expression s

~u i ~(') 5C'( ' )=u1s ~'k= l ark Oxj ~x k

and remark that also 5f" has no term of order zero. Relation (6) may now be written in the form

5C'v = - v(Y - 2m) u 1. Therefore

0-_ s 5a'(vg) N Ov 0~

= V ~ " ~ + V ~ ( " V - - 2 L k = E lU la jk 63 X j 63 Xk

N ~v c~ =--2(s k=l ~ utask ~xs ~xk

Since Re2=21, we conclude that

N Ov O~ aJk ~x---)j ex~ =0

j , k = l

in O, and hence v=const. (e.g. [3, p. 90]). Consequently 2=21. This proves the Lemma.

References

1. Hess~ P., Kato, T.: On some linear and nonIinear eigenvalue problems with an indefinite weight function. Comm. Partial Differential Equations 5, 999-1030 (1980)

2. Gossez, J.P., Lami Dozo, E.: On the principal eigenvalue of a second order linear elliptic problem. Preprint

3. Protter, M.H., Weinberger, H.F.: Maximum principles in differential equations. Englewood Cliffs, New Jersey: Prentice-Hail 1967

Received June 15, 1981