A local symmetry result for linear elliptic problems with solutions changing sign (Q719444)

It is proved that the only domain \(\Omega\) such that there exists a solution of the following overdertermined problem \[ \begin{aligned} \Delta u + \omega^2 u=-1 & \;\;\text{in }\Omega,\\ u=0 & \;\;\text{on } \partial \Omega, \\ \frac{1}{|\partial \Omega|}\int_{\partial \Omega} \partial_n u =c, & \end{aligned} \] for a given constant \(c\), is the unit ball \(B\), if one assumes that \(\Omega\) lies in an appropriate class of Lipschitz domains.