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

upon variation of , which in particular yield inequalities for the proper eigenvalues of the Dirichlet Laplacian when we further assume that the imbedding of the Sobolev space into the space is compact. In this case, we prove the same type of inequalities for the projections onto the eigenspaces upon variation of .

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