An overdetermined problem for harmonic functions
The author shows that any open bounded connected subset of \(\mathbb{R}^n\) which has real-analytic boundary and which carries \(n\) harmonic functions satisfying an appropriate partial differential equation involving the curvature at the boundary is homeomorphic to a ball. The proof uses a result by \textit{J. Serrin} [Arch. Ration. Mech. Anal. 43, 304--318 (1971; Zbl 0222.31007)] and it is based on a long equation manipulation.
scientific article; zbMATH DE number 5962673
scientific article