A note on the Schiffer conjecture
Résumé
A domain $\Omega\subset\mathbb{R}^n$ ($n\geq2$) with smooth connected boundary is said to have the Schiffer's property if there is no $\lambda>0$ such that the overdetermined boundary value problem $\Delta u + \lambda u + 1 = 0$ in $\Omega$, $u=\frac{\partial u}{\partial\nu}=0$ on $\partial\Omega$ where $\nu$ is the exterior normal to $\partial\Omega$ has a solution. We prove integral identities for the exterior normal to the boundary of domain $\Omega$ lacking the Schiffer property.