On elliptic operators with Steklov condition perturbed by Dirichlet condition on a small part of boundary

Abstract

We consider a boundary value problem for a homogeneous elliptic equation with an inhomogeneous Steklov boundary condition. The problem involves a singular perturbation, which is the Dirichlet condition imposed on a small piece of the boundary. We rewrite such problem to a resolvent equation for a self-adjoint operator in a fractional Sobolev space on the boundary of the domain. We prove the norm convergence of this operator to a limiting one associated with an unperturbed problem involving no Dirichlet condition. We also establish an order sharp estimate for the convergence rate. The established convergence implies the convergence of the spectra and spectral projectors. In the second part of the work we study perturbed eigenvalues converging to limiting simple discrete ones. We construct two-terms asymptotic expansions for such eigenvalues and for the associated eigenfunctions.

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Acknowledgements

The authors thank the referee for useful remarks, which allowed them to improve the initial version of the paper. The scientific contribution by B.D.I. to results presented in Sect. 3 was financially supported by Russian Science Foundation (Project No. 20-11-19995). The scientific contribution by C.G.A. to results presented in Sect. 2 was financially supported by Russian Science Foundation (Project No. 20-11-20272). The scientific contribution by G.C. was supported by project GNAMPA-INDAM “Analisi asintotica di problemi stazionari ed evolutivi in materiali compositi e strutture sottili” (2020).

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Borisov, D.I., Cardone, G., Chechkin, G.A. et al. On elliptic operators with Steklov condition perturbed by Dirichlet condition on a small part of boundary. Calc. Var. 60, 48 (2021). https://doi.org/10.1007/s00526-020-01847-w

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Mathematics Subject Classification

  • 35B25
  • 35P05