JournalszaaVol. 24 , No. 2DOI 10.4171/zaa/1240

A Global Lipschitz Continuity Result for a Domain Dependent Dirichlet Eigenvalue Problem for the Laplace Operator

  • Pier Domenico Lamberti

    Università di Padova, Italy
  • Massimo Lanza de Cristoforis

    Università di Padova, Italy
A Global Lipschitz Continuity Result for a Domain Dependent Dirichlet Eigenvalue Problem for the Laplace Operator cover

Abstract

Let Ω\Omega be an open connected subset of Rn{\mathbb{R}}^{n} for which the Poincar\'{e} inequality holds. We consider the Dirichlet eigenvalue problem for the Laplace operator in the open subset ϕ(Ω)\phi(\Omega) of Rn{\mathbb{R}}^{n}, where ϕ\phi is a locally Lipschitz continuous homeomorphism of Ω\Omega onto ϕ(Ω)\phi(\Omega). Then we show Lipschitz type inequalities for the reciprocals of the eigenvalues of the Rayleigh quotient

ϕ(Ω)Dv2dyϕ(Ω)v2dy\frac{\int_{\phi(\Omega)}|Dv|^{2}\,dy}{ \int_{\phi(\Omega)}|v|^{2}\,dy}

upon variation of ϕ\phi, which in particular yield inequalities for the proper eigenvalues of the Dirichlet Laplacian when we further assume that the imbedding of the Sobolev space W01,2(Ω)W^{1,2}_{0}(\Omega) into the space L2(Ω)L^{2}(\Omega) is compact. In this case, we prove the same type of inequalities for the projections onto the eigenspaces upon variation of ϕ\phi.