Serrin's result for domains with a corner or cusp (Q1974772)

The author proves that, if \(u\) is a positive solution of the differential equation \(\Delta u+1= 0\) on a bounded domain \(\Omega\subset \mathbb{R}^n\) with boundary conditions \[ u= 0,\quad {\partial u\over\partial\nu}= c\quad\text{on }\partial\Omega\setminus \{P\} \] then \(\Omega\) is a ball and \(u\) is radially symmetric. Here \(c\) is a constant, \(P\in \partial\Omega\), and \({\partial u\over\partial\nu}\) is the normal derivative along the inward normal field on \(\partial\Omega\). Here, the normal derivative is defined everywhere except at a possible corner or a cusp at \(P\). The boundary of \(\Omega\) satisfies either the interior sphere condition or the exterior sphere condition.