A concavity condition for existence of a negative Neumann-Poincar\'e eigenvalue in three dimensions

TL;DR

This paper establishes that certain concavity conditions on a 3D domain's boundary guarantee the existence of negative eigenvalues for the Neumann-Poincaré operator, linking geometric properties to spectral behavior.

Contribution

It introduces a simple concavity condition involving Gaussian curvature that ensures negative eigenvalues for the Neumann-Poincaré operator in three dimensions.

Findings

Domains with negative Gaussian curvature points have negative Neumann-Poincaré eigenvalues.

The condition applies to the boundary or its inversion in a sphere.

The concavity condition is straightforward to verify.

Abstract

It is proved that if a bounded domain in three dimensions satisfies a certain concavity condition, then the Neumann-Poincar\'e operator on the boundary of the domain or its inversion in a sphere has at least one negative eigenvalue. The concavity condition is quite simple, and is satisfied if there is a point on the boundary at which the Gaussian curvature is negative.