a1 Department of Mathematics, Sproul Hall, University of California, Riverside, CA 92521-0135, U.S.A, e-mail: firstname.lastname@example.org
a2 Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, GA 30602, U.S.A, e-mail: email@example.com
Let Ω be a non-empty open set in n with finite ‘volume’ (n-dimensional Lebesgue measure). Let be the Laplacian operator. Consider the eigenvalue problem (with Dirichlet boundary conditions):
where λ and u is a non-zero member of (the closure in the Sobolev space H1(Ω) of the set of smooth functions with compact support contained in Ω). It is well known that the values of λ for which (1·1) has a non-zero solution are positive and form a discrete set. Moreover, for each λ, the associated eigenspace is finite dimensional. Let the spectrum of (1·1) be denoted where 0 < λ1 ≤ λ2 ≤ … and where the multiplicity of each λ in the sequence is the dimension of the associated eigenspace. Let
(Received December 14 1993)
(Revised September 26 1994)
* Supported in part by NSF grants DMS-9207098 (M.L.L.) and DMS-9206784 (C.P.).