On vanishing near corners of transmission eigenfunctions

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.