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Zeitschrift für Analysis und ihre Anwendungen


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Volume 15, Issue 3, 1996, pp. 619–635
DOI: 10.4171/ZAA/719

Published online: 1996-09-30

Radial Symmetry for an Electrostatic, a Capillarity and some Fully Nonlinear Overdetermined Problems on Exterior Domains

Wolfgang Reichel[1]

(1) Karlsruhe Institute of Technology (KIT), Germany

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.

Keywords: Overdetermined problems, exterior domains, radial symmetry, electrostatic condenser, capillary surfaces

Reichel Wolfgang: Radial Symmetry for an Electrostatic, a Capillarity and some Fully Nonlinear Overdetermined Problems on Exterior Domains. Z. Anal. Anwend. 15 (1996), 619-635. doi: 10.4171/ZAA/719