Boundary localization of transmission eigenfunctions in spherically stratified media

TL;DR

This paper investigates the boundary localization of transmission eigenfunctions in spherically stratified media, showing that eigenfunctions concentrate near the boundary under certain conditions, with numerical analysis of parameter effects.

Contribution

It extends previous studies by proving boundary localization of eigenfunctions in radially symmetric media and analyzing the influence of medium parameters through numerical simulations.

Findings

Eigenfunctions concentrate near the boundary for large eigenvalues.

Existence of localized eigenfunctions in constant media.

Medium parameters affect the geometric patterns of eigenfunctions.

Abstract

Consider the transmission eigenvalue problem for $u \in H^1(\Omega)$ and $v\in H^1(\Omega)$ associated with $(\Omega; \sigma, \mathbf{n}^2)$, where $\Omega$ is a ball in $\mathbb{R}^N$, $N=2,3$. If $\sigma$ and $\mathbf{n}$ are both radially symmetric, namely they are functions of the radial parameter $r$ only, we show that there exists a sequence of transmission eigenfunctions $\{u_m, v_m\}_{m\in\mathbb{N}}$ associated with $k_m\rightarrow+\infty$ as $m\rightarrow+\infty$ such that the $L^2$-energies of $v_m$'s are concentrated around $\partial\Omega$. If $\sigma$ and $\mathbf{n}$ are both constant, we show the existence of transmission eigenfunctions $\{u_j, v_j\}_{j\in\mathbb{N}}$ such that both $u_j$ and $v_j$ are localized around $\partial\Omega$. Our results extend the recent studies in [15,16]. Through numerics, we also discuss the effects of the medium parameters, namely $\sigma$ and $\mathbf{n}$, on the geometric patterns of the transmission eigenfunctions.