On vanishing near corners of transmission eigenfunctions

Eemeli Blåsten, Hongyu LIU*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)

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=0inΩ,(Δ+k2(1+V))w=0inΩ,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.

Original languageEnglish
Pages (from-to)3616-3632
Number of pages17
JournalJournal of Functional Analysis
Volume273
Issue number11
DOIs
Publication statusPublished - 1 Dec 2017

Scopus Subject Areas

  • Analysis

User-Defined Keywords

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

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