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Abstract: We consider general second order uniformly elliptic operators subject tohomogeneous boundary conditions on open sets $\phi \Omega$ parametrized byLipschitz homeomorphisms $\phi $ defined on a fixed reference domain $\Omega$.Given two open sets $\phi \Omega$, $\tilde \phi \Omega$ we estimate thevariation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm$\|\tilde \phi -\phi \| {W^{1,p}\Omega}$ for finite values of $p$, undernatural summability conditions on eigenfunctions and their gradients. We provethat such conditions are satisfied for a wide class of operators and open sets,including open sets with Lipschitz continuous boundaries. We apply theseestimates to control the variation of the eigenvalues and eigenfunctions viathe measure of the symmetric difference of the open sets. We also discuss anapplication to the stability of solutions to the Poisson problem.



Author: G. Barbatis, V.I. Burenkov, P.D. Lamberti

Source: https://arxiv.org/



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