On vanishing near corners of transmission eigenfunctions

Eemeli Blåsten; Hongyu Liu

doi:10.1016/j.jfa.2017.08.023

paper

On vanishing near corners of transmission eigenfunctions

arXiv:1701.07957 · doi:10.1016/j.jfa.2017.08.023

Abstract

Let $Ω$ be a bounded domain in $R^{n}$ , $n \geq 2$ , and $V \in L^{\infty} (Ω)$ be a potential function. Consider the following transmission eigenvalue problem for nontrivial $v, w \in L^{2} (Ω)$ and $k \in R_{+}$ , \[(Δ+k^2)v= 0 \quad \text{in } Ω,\] \[(Δ+k^2(1+V))w= 0 \quad \text{in } Ω,\] \[w-v \in H^2_0(Ω), \quad \lVert v \rVert_{L^2(Ω)}=1. \] 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 $\partial Ω$ 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.

17 pages, addendum at arxiv:1710.08089

On vanishing near corners of transmission eigenfunctions

Abstract

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