Asymptotic expansion of solution to Dirichlet problem in perforated domain: strange term case

Abstract

We consider an elliptic operator in a multi-dimensional space periodically perforated by closely spaced small cavities. The coefficients of the differential expression are varying and infinitely differentiable functions bounded uniformly with all their derivatives. For the coefficients at higher derivatives a uniform ellipticity condition is supposed. On the boundaries of the cavities we impose the Dirichlet condition. The sizes of the cavities and the distances between them are characterized by two small parameters. They are chosen to ensure the appearance of a strange term under the homogenization, which is an additional potential in the homogenized operator. The main result of the work is the scheme for constructing two-parametric asymptotics for the resolvent of the considered operator and its application for determining the leading terms in the asymptotics. The scheme is based on a combination of the multi-scaled method and the method of matching asymptotic expansions. The former is used to take into consideration the distribution of the cavities, while the latter takes into account the geometry of the cavities and the Dirichlet condition on its boundary.