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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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##### References:
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