Regularity results for transmission problems with sign-changing coefficients: a modal approach

TL;DR

This paper analyzes scalar transmission problems involving negative materials, revealing conditions for well-posedness and regularity losses, especially highlighting the impact of geometry and interface curvature.

Contribution

It introduces a modal approach to study regularity and well-posedness in transmission problems with sign-changing coefficients, including flat interface cases.

Findings

Well-posedness with regularity loss in spherical geometries.

Potential ill-posedness with infinite-dimensional kernels in flat interfaces.

Curvature significantly influences problem regularity and solvability.

Abstract

We investigate some scalar transmission problems between a classical positive material and a negative one, whose physical coefficients are negative. First, we consider cases where the negative inclusion is a disk in 2d and a ball in 3d. Thanks to asymptotics of Bessel functions (validated numerically), we show well-posedness but with some possible loses of regularity of the solution compared to the classical case of transmission problems between two positive materials. Noticing that the curvature plays a central role, we then explore the case of flat interfaces in the context of waveguides. In this case, the transmission problem can also have some loses of regularity, or even be ill-posed (kernel of infinite dimension).