On vanishing near corners of transmission eigenfunctions

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal

7 Scopus Citations
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Author(s)

Detail(s)

Original languageEnglish
Pages (from-to)3616-3632
Journal / PublicationJournal of Functional Analysis
Volume273
Issue number11
Online published5 Sep 2017
Publication statusPublished - 1 Dec 2017
Externally publishedYes

Abstract

Let Ω be a bounded domain in Rn, n≥2, and V∈L(Ω) be a potential function. Consider the following transmission eigenvalue problem for nontrivial v,w∈L2(Ω) and k∈R+, {(Δ+k2)v=0                  in Ω, {(Δ+k2(1+V))w=0         in Ω, { w−v∈H0 2(Ω), ‖v‖L2 (Ω) =1. We show that the transmission eigenfunctions v and w carry the geometric information of supp(V). Indeed, it is proved that v and w vanish near a corner point on ∂Ω in a generic situation where the corner possesses an interior angle less than π and the potential function V does not vanish at the corner point. This is the first quantitative result concerning the intrinsic property of transmission eigenfunctions and enriches the classical spectral theory for Dirichlet/Neumann Laplacian. We also discuss its implications to inverse scattering theory and invisibility.

Research Area(s)

  • Corner, Interior transmission eigenfunction, Non-scattering, Vanishing and localizing