On vanishing near corners of transmission eigenfunctions
The paper investigates the following (interior) transmission eigenvalue problem for nontrivial \(v, w\in L^2(\Omega)\), \[ \begin{cases} (\Delta+k^2)v=0\quad &\text{in}\;\;\Omega,\\ (\Delta+k^2(1+V))w=0\quad &\text{in}\;\;\Omega,\\ w-v\in H^2_0(\Omega),\;\;||v||_{L^2(\Omega)}&=1, \end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(n\geq2\), \(k\in \mathbb{R}_+\) and \(V\in L^\infty(\Omega)\) is a potential function. The transmission eigenvalue problem is a type of non elliptic and non self-adjoint problem. The existing results are mainly concerned with the spectral properties of the transmission eigenvalues, including the existence, discreteness and infiniteness, and Weyl laws. In this paper the vanishing properties of interior transmission eigenfunctions is investigated. It is proved that \(v\) and \(w\) vanish near a corner point on \(\partial \Omega\) 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 proved by an indirect approach, connecting to the wave scattering theory. Note that the vanishing behavior carries geometric information of the support of the involved potential function \(V\). Implications of the obtained resuls to inverse scattering theory and invisibility are also discussed at end of the paper.
scientific article