Upper semicontinuity of eigenvalues of selfadjoint operators defined on moving domains
Satoshi Kaizu · Proceedings of the Japan Academy Series A Mathematical Sciences · 1985
Introduction.We are interested in "wild" perturbations in the sense of J. Rauch and M. Taylor [6], on eigenvalue problems for the Laplacian.We show the upper semicontinuity of each k-th eigenvalue of the minus Laplacian with respect to a domain perturbation belonging to a certain class.This class contains a perturbation argued by the author [5].Hereafter we describe all statements only in an abstract fashion.Let X and V, be real, separable and infinitely dimensional Hilbert spaces with X V,.We assume that the injection V,-,X is compact.We denote by and (,) the norm and inner product on X, respectively.Here, means the value zero or the values of a sequence decreasing to zero.Let a," V, V,-R be a symmetric continuous bilinear form such that a,(v)e, llvll for all v e V,, where a,(v)=a,(v, v) and c, is a positive constant.We denote by H, the closure of V, in X and denote by P, the orthogonal projection from X onto H,.We set 2={x eX] Ix I=1).We define a positive selfadjoint operator A," D(A,)--.H, by a,(u, v)=(A,u, v) for all u e D(A,) and v e V,, where D(A,)={u e V,lc>0 such that <=c [v] for all v e V,).We consider the equation" A,u,=/,u,, /, e R and u, e X. Let/ be the k-th eigenvalue of A, counting with its multiplicity;