Affiliations: [a] Department of Mathematics, Jilin University, Changchun, Jilin, China | [b] Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China | [c] School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, Jiangsu, China
Abstract: 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.
