Homogenization of the Poisson equation in a non-periodically perforated domain

Article type: Research Article

Authors: Blanc, Xavier^{; *} | Wolf, Sylvain

Affiliations: Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, France. E-mails: [email protected], [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of the perforations is denoted by ε>0, and is proportional to the distance between neighbouring perforations. In the periodic case, the homogenized problem (obtained in the limit ε→0) is well understood (see (Rocky Mountain J. Math. 10 (1980) 125–140)). We extend these results to a non-periodic case which is defined as a localized deformation of the periodic setting. We propose geometric assumptions that make precise this setting, and we prove results which extend those of the periodic case: existence of a corrector, convergence to the homogenized problem, and two-scale expansion.

Keywords: Homogenization, Perforated domain, Poisson equation

DOI: 10.3233/ASY-201667

Journal: Asymptotic Analysis, vol. 126, no. 1-2, pp. 129-155, 2022