Radial symmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains
Summary: We consider two physically motivated problems: (1) Suppose the surface of a body in \(\mathbb{R}^2\) or \(\mathbb{R}^3\) is charged with a constant density. If the induced single-layer potential is constant inside the body, does it have to be a ball? (2) Suppose a straight solid cylinder of unknown cross-section is dipped into a large plain liquid reservoir. If the liquid rises to the same height on the cylinder wall, does the cylinder necessarily have circular cross-section? Both questions are answered with yes, and both problems are shown to be of the type \[ \text{div} (g(|\nabla u|)\nabla u)+f(u,|\nabla u|)=0\text{ in }\Omega,\;u=\text{const},\;{{\partial u}\over{\partial\nu}}= \text{const on }\partial\Omega,\;u=0\text{ at }\infty, \] where \(\partial_u f\leq 0\) and \(\Omega=\mathbb{R}^N \setminus\overline{G}\) is the connected exterior of the smooth bounded domain \(G\). The overdetermined nature of this possibly degenerate boundary value problem forces \(\Omega\) to be radial. This is shown by a variant of the Alexandroff-Serrin method of moving hyperplanes. The results extend to Monge-Ampère equations.
scientific article; zbMATH DE number 935175