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 ): ∇ · ( σ ∇ u ) + k 2 n 2 u = 0 in Ω , Δ v + k 2 v = 0 in Ω , u = v , σ ∂ u ∂ ν = ∂ v ∂ ν on ∂ 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 { u m , v m } m ∈ N associated with k m → + ∞ as m → + ∞ such that the L 2 -energies of v m ’s are concentrated around ∂ Ω. If σ and n are both constant, we show the existence of transmission eigenfunctions { u j , v j } j ∈ N such that both u j and v j are localized around ∂ Ω. Our results extend the recent studies in (SIAM J. Imaging Sci. 14 (2021), 946–975; Chow et al.). Through numerics, we also discuss the effects of the medium parameters, namely σ and n, on the geometric patterns of the transmission eigenfunctions.