On vanishing near corners of conductive transmission eigenfunctions

TL;DR

This paper investigates the geometric properties of transmission eigenfunctions in conductive problems, proving they vanish near boundary corners under certain regularity conditions, which enhances understanding of wave behavior in such media.

Contribution

It establishes that transmission eigenfunctions generally vanish around boundary corners, extending previous results and providing new insights into their geometric structure under mild regularity assumptions.

Findings

Eigenfunctions vanish near boundary corners under mild regularity conditions

Growth rate of Herglotz transform characterizes eigenfunction regularity

Vanishing near corners is a generic local geometric property

Abstract

In this paper, we consider the transmission eigenvalue problem associated with a general conductive transmission condition and study the geometric structures of the transmission eigenfunctions. We prove that under a mild regularity condition in terms of the Herglotz approximations of one of the pair of the transmission eigenfunctions, the eigenfunctions must be vanishing around a corner on the boundary. The Herglotz approximation can be regarded as the Fourier transform of the transmission eigenfunction in terms of the plane waves, and the growth rate of the transformed function can be used to characterize the regularity of the underlying wave function. The geometric structures derived in this paper include the related results in [5,19] as special cases and verify that the vanishing around corners is a generic local geometric property of the transmission eigenfunctions.