Abstract
We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets ø(Ω) parametrized by Lipschitz homeomorphisms ø defined on a fixed reference domain Ω. For two open sets ø(Ω) and eø(Ω) 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 eigen-functions 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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Barbatis, G., Burenkov, V.I., Lamberti, P.D. (2010). Stability Estimates for Resolvents, Eigenvalues, and Eigenfunctions of Elliptic Operators on Variable Domains. In: Laptev, A. (eds) Around the Research of Vladimir Maz'ya II. International Mathematical Series, vol 12. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1343-2_2
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DOI: https://doi.org/10.1007/978-1-4419-1343-2_2
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