Boundary localization of transmission eigenfunctions in spherically stratified media

摘要

Consider the transmission eigenvalue problem for u. H1( Omega) and v. H1( Omega):...... center dot (s. u) + k2n2u = 0 in , del.(sigma del u) + k(2) n(2)u = 0 in Omega, Delta u k(2) v, =.v.. . Omega, where is a ball in RN, N = 2, 3. If s 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. +8 as m.+8 such that the L2-energies of vm's are concentrated around.. If s 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. . 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 s and n, on the geometric patterns of the transmission eigenfunctions.