Boundary localization of transmission eigenfunctions in spherically stratified media

Abstract

Consider the transmission eigenvalue problem for u ∈ H 1 (Ω) and v ∈ H 1 (Ω):where Ω is a ball in R N , N = 2, 3. If σ and 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 {um, vm} m∈N associated with km → +∞ as m → +∞ such that the L 2 -energies of vm's are concentrated around ∂Ω. If σ and n are both constant, we show the existence of transmission eigenfunctions {uj, vj} j∈N such that both uj and vj are localized around ∂Ω. Our results extend the recent studies in [15,16]. Through numerics, we also discuss the effects of the medium parameters, namely σ and n, on the geometric patterns of the transmission eigenfunctions.