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

Pu-Zhao Kow, Mikko Salo, Henrik Shahgholian

TL;DR

The paper develops a free-boundary (Bernoulli) framework to study scattering by anisotropic inhomogeneous media with corner and edge singularities. Through blowup analysis and a balanced energy functional, it shows that nontrivial scattering occurs at corners/edges under precise nondegeneracy and irrational-angle conditions, while certain degenerate or symmetric cases can admit invisibility. The results extend previous isotropic/nonscattering analyses to general divergence-form operators, providing a systematic approach to classify when geometric singularities enforce scattering at fixed frequencies. The methods yield both regularity insights for the associated free boundary and sharp criteria distinguishing scattering from non-scattering configurations. Overall, the work deepens understanding of how anisotropy, contrast, and boundary geometry govern wave propagation and scattering in complex media.

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 $\notinπ\mathbb{Q}$ always produce nontrivial scattering for any incoming wave.

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

TL;DR

The paper develops a free-boundary (Bernoulli) framework to study scattering by anisotropic inhomogeneous media with corner and edge singularities. Through blowup analysis and a balanced energy functional, it shows that nontrivial scattering occurs at corners/edges under precise nondegeneracy and irrational-angle conditions, while certain degenerate or symmetric cases can admit invisibility. The results extend previous isotropic/nonscattering analyses to general divergence-form operators, providing a systematic approach to classify when geometric singularities enforce scattering at fixed frequencies. The methods yield both regularity insights for the associated free boundary and sharp criteria distinguishing scattering from non-scattering configurations. Overall, the work deepens understanding of how anisotropy, contrast, and boundary geometry govern wave propagation and scattering in complex media.

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 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 obstacles with corners in two dimensions (and with edge points in higher dimensions) with angles always produce nontrivial scattering for any incoming wave.

Paper Structure

This paper contains 13 sections, 16 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Definitions and terminology
  4. Methodology
  5. Statements of the main results and related work
  6. Vanishing Bernoulli condition
  7. Nonvanishing Bernoulli condition
  8. Proof of Theorems \ref{['thm:1']} and \ref{['thm:2']}
  9. Lipschitz regularity and optimal decay rate
  10. Blowup solutions
  11. Non-degeneracy and weak flatness
  12. Conclusions
  13. Proof of Theorems \ref{['thm:3']} and \ref{['thm:4']}

Key Result

Theorem 1.4

Let $D$ be a bounded open set in $\mathbb{R}^{2}$, and suppose that Let $\kappa>0$, let $\rho\in L^{\infty}(D)$ be a positive real-valued function, and let $A\in (C^{1} (\overline{D}))^{n\times n}$ be a real symmetric matrix-valued function, satisfying the condition of uniform ellipticity eq:ellipticity. Suppose that there exists $x_{0}\in\partial D$ such that If $\partial D$ is not $C^{1}$ near

Theorems & Definitions (26)

  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Example 1.5: Necessity of condition \ref{['eq:non-degeneracy']}
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 16 more