Spectral stability for a class of fourth order Steklov problems under domain perturbations
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We study the spectral stability of two fourth order Steklov problems upon domain perturbation. One of the two problems is the classical DBS—Dirichlet Biharmonic Steklov—problem, the other one is a variant. Under a comparatively weak condition on the convergence of the domains, we prove the stability of the resolvent operators for both problems, which implies the stability of eigenvalues and eigenfunctions. The stability estimates for the eigenfunctions are expressed in terms of the strong \(H^2\)-norms. The analysis is carried out without assuming that the domains are star-shaped. Our condition turns out to be sharp at least for the variant of the DBS problem. In the case of the DBS problem, we prove stability of a suitable Dirichlet-to-Neumann type map under very weak conditions on the convergence of the domains and we formulate an open problem. As bypass product of our analysis, we provide some stability and instability results for Navier and Navier-type boundary value problems for the biharmonic operator.
Mathematics Subject Classification35J40 35B20 35P15
The authors are grateful to the anonymous referee for the very careful reading of the paper and for the many useful comments and suggestions which helped them to improve the presentation of the paper, and for bringing to their attention item  in the references list. The authors are also very thankful to Professors José M. Arrieta and Giles Auchmuty for useful discussions and references.
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