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∈H02(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.
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