Operator estimates for the crushed ice problem

Article type: Research Article

Authors: Khrabustovskyi, Andrii^{a; *} | Post, Olaf^b

Affiliations: [a] Institut für Angewandte Mathematik, Technische Universität Graz, 8010 Graz, Austria. E-mail: khrabustovskyi@math.tugraz.at | [b] Fachbereich 4 – Mathematik, Universität Trier, 54286 Trier, Germany. E-mail: olaf.post@uni-trier.de

Correspondence: [*] Corresponding author. E-mail: khrabustovskyi@math.tugraz.at.

Abstract: Let ΔΩε be the Dirichlet Laplacian in the domain Ωε:=Ω∖(⋃iDiε). Here Ω⊂Rn and {Diε}i is a family of tiny identical holes (“ice pieces”) distributed periodically in Rn with period ε. We denote by cap(Diε) the capacity of a single hole. It was known for a long time that −ΔΩε converges to the operator −ΔΩ+q in strong resolvent sense provided the limit q:=limε→0cap(Diε)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 kth eigenvalue of −ΔΩε and −ΔΩε+q. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.

Keywords: Crushed ice problem, homogenization, norm resolvent convergence, operator estimates, varying Hilbert spaces

DOI: 10.3233/ASY-181480

Journal: Asymptotic Analysis, vol. 110, no. 3-4, pp. 137-161, 2018