Boundary localization of transmission eigenfunctions in spherically stratified media

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 $k_m\to +\infty$ as $m\to +\infty$ 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.