Transmission eigenvalues and a problem of Hans Lewy (Q1570984)

Under certain assumptions on the matrix \(N= N(x)\), the authors consider the transmission eigenvalue problem \[ \nabla\cdot N\nabla u+ k^2u= 0\quad\text{and}\quad \Delta v+ k^2v= 0\quad\text{in}\quad D \] \[ \text{together with}\quad u= v\quad\text{and}\quad \partial u/\partial\nu= \partial v/\partial\nu\quad\text{on}\quad \partial D. \] Here, \(D\subset\mathbb{R}^n\) denotes a bounded and simply connected region with smooth boundary \(\partial D\) and unit normal \(\nu\). A main difficulty is due to the fact that \(N(x)- I\) vanishes at \(\partial D\). Establishing in particular an associated weigthed Poincaré inequality, it is shown that the problem exhibits at most a countable set of transmission eigenvalues \(k\geq 0\). The result has important applications to the inverse scattering problem for anisotropic media.