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Symmetry for exterior elliptic problems and two conjectures in potential theory. (English) Zbl 0997.35014
Summary: The author extends a classical result of Serrin to a class of elliptic problems \(\Delta u + f(u, |\nabla u|) = 0\) in exterior domains \(\mathbb{R}^N\setminus G\) (or \(\Omega\setminus G\) with \(\Omega\) and \(G\) bounded). In case \(G\) is an union of a finite number of disjoint \(C^2\)-domains \(G_i\) and \(u = a_i > 0\), \(\partial u/\partial n=\alpha_i\leq 0\) on \(\partial G_i\), \(u\to 0\) at infinity, he shows that if a non-negative solution of such a problem exists, then \(G\) has only one component and it is a ball. As a consequence he establishes two results in electrostatics and capillarity theory. He further obtains symmetry results for quasilinear elliptic equations in the exterior of a ball.

MSC:
35J45 Systems of elliptic equations, general (MSC2000)
35Q72 Other PDE from mechanics (MSC2000)
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
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