On vanishing near corners of transmission eigenfunctions

Let $\mathrm{\Omega }$ be a bounded domain in ${\mathbb{R}}^n$, $n\ge 2$, and $V\in L^\infty(\Omega)$ be a potential function. Consider the following transmission eigenvalue problem for nontrivial $v,w\in L^2(\mathrm{\Omega })$ and $k\in {\mathbb{R}}_{+}$, $$(\mathrm{\Delta }+k^2)v=0\text{in }\mathrm{\Omega },$$ $$(\mathrm{\Delta }+k^2(1+V))w=0\text{in }\mathrm{\Omega },$$ $$w-v\in H_0^2(\mathrm{\Omega }),\parallel v{\parallel }_{L^2(\mathrm{\Omega })}=1.$$ We show that the transmission eigenfunctions $v$ and $w$ carry the geometric information of $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(V)$. Indeed, it is proved that $v$ and $w$ vanish near a corner point on $\mathrm{\partial }\mathrm{\Omega }$ in a generic situation where the corner possesses an interior angle less than $\pi$ 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.