Norm-Resolvent Convergence in Perforated Domains

Abstract

For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator $-\Delta$ in the perforated domain $\Omega \setminus {\bigcup }_{i\in 2\epsilon {\mathbb{Z}}^d}{B}_{a_{\epsilon }}(i),$ $a_{\epsilon }\ll \epsilon ,$ 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.

18 pages, 2 figures