# Operator estimates for the crushed ice problem

Khrabustovskyi, Andrii; Post, Olaf

Abstract:
Let Δ$_{Ωε}$ be the Dirichlet Laplacian in the domain Ωε := Ω \ (∪$_{i}$D$_{iε}$). Here Ω ⊂ R$^{n}$and {D$_{iε}$}$_{i}$ is a family of tiny identical holes (“ice pieces”) distributed periodically in R$^{n}$ with period ε. We denote by cap (D$_{iε}$) the capacity of a single hole. It was known for a long time that −Δ$_{Ωε}$ converges to the operator −Δ$_{Ω}$ $+$ $q$ in strong resolvent sense provided the limit $q$ : $=$ lime$_{ε→0}$→0 cap(D$_{iε}$)ε$^{-n}$ exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded Ω) an estimate for the difference of the $k$-th eigenvalue of −Δ$_{Ωε}$ and −Δ$_{Ωε}$ $+$ $q$. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2017 Sprache Englisch Identifikator ISSN: 2365-662X URN: urn:nbn:de:swb:90-756565 KITopen-ID: 1000075656 Verlag KIT, Karlsruhe Umfang 22 S. Serie CRC 1173 ; 2017/24 Schlagworte crushed ice problem, homogenization, norm resolvent convergence, operator estimates, varying Hilbert spaces
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