Interactions between moderately close inclusions for the 2D Dirichlet-Laplacian

TL;DR

This paper extends the understanding of the asymptotic behavior of solutions to the Dirichlet-Laplace problem in 2D domains with small, arbitrarily shaped inclusions, considering multiple scales and less regular data.

Contribution

It generalizes previous results to inclusions of any shape, relaxes regularity assumptions, and analyzes multi-scale configurations using conformal mapping techniques.

Findings

Asymptotic expansions for general-shaped inclusions

Analysis of multiple scale configurations

Relaxed regularity and support assumptions

Abstract

This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension greater than two or Dirichlet condition in dimension greater than three. The case of two circular inclusions in a bidimensional domain was considered in [1]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions. We also analyze configurations with several scales for the distance between the inclusions (when the number is larger than 2).