# 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

## Abstract

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

upon variation of $ϕ$, which in particular yield inequalities for the proper eigenvalues of the Dirichlet Laplacian when we further assume that the imbedding of the Sobolev space $W_{0}(Ω)$ into the space $L_{2}(Ω)$ is compact. In this case, we prove the same type of inequalities for the projections onto the eigenspaces upon variation of $ϕ$.

## Cite this article

Pier Domenico Lamberti, Massimo Lanza de Cristoforis, A Global Lipschitz Continuity Result for a Domain Dependent Dirichlet Eigenvalue Problem for the Laplace Operator. Z. Anal. Anwend. 24 (2005), no. 2, pp. 277–304

DOI 10.4171/ZAA/1240