Norm-resolvent convergence in perforated domains

Kirill Cherednichenko, Patrick Dondl, Frank Rösler

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Abstract

For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator Δ in the perforated domain Ω\∪ i∈2ϵℤd B (i), a ϵ ϵ, to the limit operator -Δ+ μ i on L 2(Ω), where μ i ∈ ℂ is a constant depending on the choice of boundary conditions. This is an improvement of previous results [Progress in Nonlinear Differential Equations and Their Applications 31 (1997), 45-93; in: Proc. Japan Acad., 1985], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem.

Original languageEnglish
Pages (from-to)163-184
Number of pages22
JournalAsymptotic Analysis
Volume110
Issue number3-4
DOIs
Publication statusPublished - 6 Dec 2018

Keywords

  • Analysis of PDE
  • Homogenisation
  • Norm-resolvent convergence
  • Perforated domain

ASJC Scopus subject areas

  • Mathematics(all)

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