Boundary localization of transmission eigenfunctions in spherically stratified media

Consider the transmission eigenvalue problem for u∈H1(Ω) and v∈H1(Ω): ∇·(σ∇u)+k2n2u=0in Ω,Δv+k2v=0in Ω,u=v,σ∂u∂ν=∂v∂νon ∂Ω, where Ω is a ball in RN, 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 L2-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 (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.