Wildly perturbed manifolds: norm resolvent and spectral convergence

Abstract

The publication of the important work of Rauch and Taylor [J. Funct. Anal. 18 (1975)] started a hole branch of research on wild perturbations of the Laplace–Beltrami operator. Here, we extend certain results and show norm convergence of the resolvent. We consider a (not necessarily compact) manifold with many small balls removed, the number of balls can increase as the radius is shrinking, the number of balls can also be infinite. If the distance of the balls shrinks less fast than the radius, then we show that the Neumann Laplacian converges to the unperturbed Laplacian, i.e., the obstacles vanish. In the Dirichlet case, we consider two cases here: if the balls are too sparse, the limit operator is again the unperturbed one, while if the balls concentrate at a certain region (they become “solid” there), the limit operator is the Dirichlet Laplacian on the complement of the solid region. Norm resolvent convergence in the limit case of homogenisation is treated by Khrabustovskyi and the second author in another article (see also the references therein). Our work is based on a norm convergence result for operators acting in varying Hilbert spaces described in a book from 2012 by the second author.