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Zeitschrift für Analysis und ihre Anwendungen


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Volume 24, Issue 2, 2005, pp. 277–304
DOI: 10.4171/ZAA/1240

Published online: 2005-06-30

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

Pier Domenico Lamberti[1] and Massimo Lanza de Cristoforis[2]

(1) Università di Padova, Italy
(2) Università di Padova, Italy

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 $$ \frac{\int_{\phi(\Omega)}|Dv|^{2}\,dy}{ \int_{\phi(\Omega)}|v|^{2}\,dy} $$ 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 $\phi$.

Keywords: Dirichlet eigenvalues and eigenvectors, Laplace operator, domain perturbation, special nonlinear operators

Lamberti Pier Domenico, Lanza de Cristoforis Massimo: A Global Lipschitz Continuity Result for a Domain Dependent Dirichlet Eigenvalue Problem for the Laplace Operator. Z. Anal. Anwend. 24 (2005), 277-304. doi: 10.4171/ZAA/1240