Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients

Details Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients

2011-01-13

arxiv.org/abs/1101.2545v1

Mathematics

Analysis of PDEs

25 pages

Scientific paper

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $\Omega $ in ${\mathbb{R}}^N$. We consider deformations $\phi (\Omega)$ of $\Omega $ obtained by means of a locally Lipschitz homeomorphism $\phi $ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of $\phi $. We prove general stability estimates without using uniform upper bounds for the gradients of the maps $\phi$. As an application, we obtain estimates on the rate of convergence for eigenvalues and eigenfunctions when a domain with an outward cusp is approximated by a sequence of Lipschitz domains.

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