Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains

Details Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains

2008-10-21

arxiv.org/abs/0810.3823v2

Mathematics

Analysis of PDEs

34 pages. Minor changes in the introduction and the refercenes. Published in: Around the research of Vladimir Maz'ya II, pp23-

Scientific paper

10.1007/978-1-4419-1343-2_2

We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by Lipschitz homeomorphisms $\phi $ defined on a fixed reference domain $\Omega$. Given two open sets $\phi (\Omega)$, $\tilde \phi (\Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\|\tilde \phi -\phi \|_{W^{1,p}(\Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.

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