An inverse problem for an elliptic equation with an affine term (Q1568703)

The author proves that for a large class of planar domains \(\Omega\) (including all true ellipses) and any real-valued \(\psi\in C^\infty (\partial \Omega)\) which is not identically constant there exist at most \(k_0\) different pairs of coefficients \((\lambda_k,\mu_k) \in\mathbb{R}^2\) such that the Cauchy problem \[ \Delta u=-\lambda_k u-\mu_k\text{ in }\Omega,\quad u=0, {\partial u\over \partial n}=\psi\text{ on }\partial\Omega \] has a solution. The numbers \(k_0\), \(\lambda_k\), and \(\mu_k\) are explicitly computable.