A free boundary approach to non-scattering obstacles with vanishing contrast

TL;DR

This paper introduces free boundary methods for obstacle problems with sign-changing solutions, revealing that certain obstacle geometries always produce nontrivial scattering in inverse problems.

Contribution

It develops novel free boundary results for obstacle problems with variable sign and relates the right hand side to harmonic polynomials, advancing inverse scattering theory.

Findings

Piecewise $C^1$ or convex obstacles in 2D cause nontrivial scattering.

Edge points in higher dimensions also cause nontrivial scattering.

New free boundary results not previously available in literature.

Abstract

Motivated by questions in inverse scattering theory, we develop free boundary methods in obstacle problems where both the solution and the right hand side of the equation may have varying sign. The key condition that prevents the appearance of corners is that the right hand side should be related to a harmonic polynomial. In this setting we prove new free boundary results not found in existing literature. Notably, our results imply that piecewise $C^1$ or convex penetrable obstacles in two dimensions and edge points in higher dimensions always cause nontrivial scattering of any incoming wave.