Asymptotic treatment of perforated domains without homogenization

By Vladimir G. Maz'ya, V. Maz’ya, Alexander Movchan. 2009

62 citations recorded by the source corpus.

As a main result of the paper, we construct and justify an asymptotic approximation of Green's function in a domain with many small inclusions. Periodicity ofthe array ofinclusions is not required. We start with an analysis of the Dirichlet problem for the Laplacian in such a domain to illustrate a method of mesoscale asymptotic approximations for solutions of boundary value problems in multiply perforated domains. The asymptotic formula obtained involves a linear combination of solutions to certain model problems whose coefficients satisfy a linear algebraic system. The solvability of this system is proved under weak geometrical assumptions, and both uniform and energy estimates for the remainder term are derived.

Subfields: Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/421fb1c6607e8781608d092fbb66ad93c605fa49