Norm-Resolvent Convergence in Perforated Domains

TL;DR

This paper proves norm-resolvent convergence of the Laplacian in perforated domains with various boundary conditions to a limit operator, improving previous results by establishing stronger convergence and spectral Hausdorff convergence.

Contribution

It establishes norm-resolvent convergence for the Laplacian in perforated domains under Dirichlet, Neumann, and Robin conditions, extending and strengthening prior results.

Findings

Proves norm-resolvent convergence for multiple boundary conditions.

Shows Hausdorff convergence of the spectrum.

Improves upon previous strong resolvent convergence results.

Abstract

For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator $-\Delta$ in the perforated domain $\Omega\setminus \bigcup_{ i\in 2\varepsilon\mathbb Z^d }B_{a_\varepsilon}(i),$ $a_\varepsilon\ll\varepsilon,$ to the limit operator $-\Delta+\mu_{\iota}$ on $L^2(\Omega)$, where $\mu_\iota\in\mathbb C$ 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.