Radial Symmetry for an Electrostatic, a Capillarity and some Fully Nonlinear Overdetermined Problems on Exterior Domains
Wolfgang Reichel · Zeitschrift für Analysis und ihre Anwendungen · 1996
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 \mathrm {div} (g(|\triangledown u |)\triangledown u) + f(u, | \triangledown u|) = 0 \ \mathrm {in} \ \Omega, \ \ u = \mathrm {const}, \frac{\partial u}{\partial v} = \mathrm {const} \ \mathrm {on} \ \partial \Omega, \ u = 0 \ \mathrm {at} \ \infty where \partial_u f ≤ 0 and \Omega = \mathbb R^N \ \bar{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, as recently developed for exterior domains by the author in [19]. The results extend to Monge-Ampere equations.