On the growth properties of interior transmission eigenfunctions near corners

TL;DR

This paper proves that interior transmission eigenfunctions localize at non-convex corners and vanish at convex corners, providing new theoretical insights into their behavior near corners with less smoothness assumptions.

Contribution

It offers the first rigorous proof of eigenfunction localization at non-convex corners and establishes vanishing at convex corners under weaker smoothness conditions.

Findings

Eigenfunctions localize at non-convex corners

Eigenfunctions vanish at convex corners

Reduced smoothness assumptions for vanishing results

Abstract

We investigate the localization and vanishing of $L^2$ interior transmission eigenfunctions at corners. Past numerical computations suggest that these eigenfunctions localize at non-convex corners. This phenomenon has, however, not been proven theoretically. We show that localization does indeed occur for some eigenfunctions at a non-convex corner. We also investigate the vanishing of interior transmission eigenfunctions at a convex corner. We prove that these eigenfunctions vanish at convex corners with reduced smoothness assumptions compared to earlier results.