Operator estimates for homogenization of the Robin Laplacian in a perforated domain

0561934 - ÚJF 2023 RIV US eng J - Journal Article
Khrabustovskyi, Andrii - Plum, M.
Operator estimates for homogenization of the Robin Laplacian in a perforated domain.
Journal of Differential Equations. Roč. 338, NOV (2022), s. 474-517. ISSN 0022-0396. E-ISSN 1090-2732
R&D Projects: GA ČR(CZ) GA21-07129S
Institutional support: RVO:61389005
Keywords : Homogenization * Perforated domain * Norm resolvent convergence * Operator estimates * Spectral convergence * Varying Hilbert spaces
OECD category: Pure mathematics
Impact factor: 2.4, year: 2022
Method of publishing: Limited access
https://doi.org/10.1016/j.jde.2022.08.005

Let epsilon > 0 be a small parameter. We consider the domain omega := omega\ omega epsilon, where omega is an open domain in Rn, and D epsilon is a family of small balls of the radius d epsilon = o(epsilon) distributed periodically with period epsilon. Let ?epsilon be the Laplace operator in ?epsilon subject to the Robin condition partial differential u partial differential n + gamma epsilon u = 0 with gamma epsilon <= 0 on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have shown that, under appropriate assumptions on d epsilon and gamma epsilon, the operator ?epsilon converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in omega and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of L2 -> L2 and L2 -> H1 operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators.(c) 2022 Elsevier Inc. All rights reserved.
Permanent Link: https://hdl.handle.net/11104/0334363