Stability Estimates for Resolvents, Eigenvalues, and Eigenfunctions of Elliptic Operators on Variable Domains

  • Gerassimos Barbatis
  • Victor I. Burenkov
  • Pier Domenico Lamberti
Part of the International Mathematical Series book series (IMAT, volume 12)


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.


Elliptic Operator Neumann Boundary Condition Stability Estimate Spectral Stability Poisson Problem 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Gerassimos Barbatis
    • 1
  • Victor I. Burenkov
    • 2
  • Pier Domenico Lamberti
    • 2
  1. 1.Department of MathematicsUniversity of AthensAthensGreece
  2. 2.Dipartimento di Matematica Pura ed ApplicataUniversità degli Studi di PadovaPadovaItaly

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