Norm-Resolvent Convergence in Perforated Domains

2017/06/19 by Patrick Dondl, Dondl, Patrick, Kirill Cherednichenko +3 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1706.05859

openalex publication_date 2017/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator -Δ in the perforated domain Ω∖ \bigcup _{i∈ 2ε\mathbb Z^d} B_aε(i), a_ε≪ε, to the limit operator -Δ+μ_ι on L²(Ω), where μ_ι∈\mathbb C is a constant depending on the choice of boundary conditions. This is an improvement of previous results [Cioranescu & Murat. A Strange Term Coming From Nowhere, Progress in Nonlinear Differential Equations and Their Applications, 31, (1997)], [S. Kaizu. The Robin Problems on Domains with Many Tiny Holes. Pro c. Japan Acad., 61, Ser. A (1985)], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem.

Cited by

• An indefinite Coulomb interaction from the Steklov spectrum of perforated manifolds
• A not-so-strange term coming from somewhere

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