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

Abstract

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

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