The spectrum of the Laplacian and boundary perturbation, I
Daisuke Fujiwara, Masao Tanikawa, Shûichi Yukita · Proceedings of the Japan Academy Series A Mathematical Sciences · 1978
Introduction.Let D be a bounded domain in R with smooth boundary ,.We assume, for simplicity, that t2 is simply connected.Consider eigenvalue problem or the Laplacian under Dirichlet con- dition ( 1{(-zl-)u(x) =0 x e 9 Let 0_<_<_<.be the eigenvalues o the problem (1).These are functions of ,.The totality of the boundaries ,, with appropriate regularity, forms a separable Hilbert manifold F. A subset of F is called residual if it is a countable intersection of open dense subsets of F. Our main theorem is Theorem 1, There is a residual subset B of F such that for any e B all the eigenspaces of the problem (1) are of dimension one.Since the complement of B is a set o first category, Theorem 1 means that or generic ,the eigenvalues of the Laplacian are all simple.Similar results were already obtained by Uhlenbeck [4] in the case of potential p.erturbation or in the case that 1 is the Laplace Beltrami operator on a compact Riemannian manifold and the perturbation is that of the metric.Theorem 1 was conjectured by Arnold [1].But the proo was not given as far as the present authors know.Our proof can easily be generalized to the case that/2 is a domain of R.1.The Hilbert manifold T' of boundary curves.Let S be the unit circle--{e 0_ 1) be the totality of embeddings