An integral equation method for the Helmholtz problem in the presence of small anisotropic inclusions

Abstract

We consider the Helmholtz problem with source term in an anisotropic domain of ${\mathbb{R}}^3$. The aim of this paper is to investigate the interplay between the geometry and analysis of elliptic equations under small perturbation of domain. The solving of this problem, anisotropic as well as isotropic case, is based on integral equations. We exhibit the Lippmann-Schwinger integral equation in the presence of finite number of anisotropic inclusions of small diameter. We derive some results for convergence estimates.

Keywords:

• Helmholtz problem
• anisotropic inclusions
• integral operators
• asymptotic expansion

AMS SUBJECT CLASSIFICATIONS:

• 35C15
• 35J05
• 35J08
• 47A55
• 78A45

Acknowledgments

The authors would like to record sincere gratitude to the editor and the anonymous referees for their interest in this research paper and their useful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).