We consider two physically motivated problems: (1) Suppose the surface of a body in or 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
where and is the connected exterior of the smooth bounded domain . The overdetermined nature of this possibly degenerate boundary value problem forces 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 . The results extend to Monge-Ampere equations.
Cite this article
Wolfgang Reichel, Radial Symmetry for an Electrostatic, a Capillarity and some Fully Nonlinear Overdetermined Problems on Exterior Domains. Z. Anal. Anwend. 15 (1996), no. 3, pp. 619–635DOI 10.4171/ZAA/719