On scattering behavior of corner domains with anisotropic inhomogeneities: part II

TL;DR

This paper investigates how anisotropic inhomogeneous corner domains scatter waves, establishing conditions under which scattering is always nontrivial, especially focusing on geometric features like corners and edges with irrational angles.

Contribution

It extends previous work by analyzing scattering in anisotropic inhomogeneous media, proving nontrivial scattering for obstacles with corners and edges under specific geometric conditions.

Findings

Corners and edges always induce nontrivial scattering in 2D and higher dimensions.

Obstacles with corners having angles not in πQ always produce nontrivial scattering.

The study uses free boundary techniques to analyze scattering behavior.

Abstract

We study the scattering behavior of an anisotropic inhomogeneous Lipschitz medium at a fixed wave number, continuing our previous work [SIAM J. Math. Anal., 56(4):4834-4853, 2024] and using free boundary techniques from [arXiv:2506.22328]. Our main results can be categorized into two distinct cases. In the first case, we show that in two dimensions, piecewise $C^{1}$ or convex penetrable obstacles with corners, and in higher dimensions, obstacles with edge points, always induce nontrivial scattering for any incoming wave. In the second case, we prove that piecewise $C^{1}$ obstacles with corners in two dimensions (and with edge points in higher dimensions) with angles $\notin\pi\mathbb{Q}$ always produce nontrivial scattering for any incoming wave.