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Radial symmetry for elliptic boundary-value problems on exterior domains. (English) Zbl 0891.35006

A special case of a well-known theorem of symmetry for elliptic equations due to J. Serrin [Arch. Ration. Mech. Anal. 43, 304-318 (1971; Zbl 0222.31007)] reads as follows: if \(\Omega\) is a bounded \(C^2\) domain in \(\mathbb{R}^N\) and \(u\in C^2(\overline\Omega)\) is a solution of \[ \Delta u+ f(u,|\nabla u|)= 0\quad\text{and} \quad u>0\text{ in } \Omega, \]
\[ u= 0\quad\text{and} \quad{\partial u\over\partial\nu}=\text{constant}\quad\text{on } \partial\Omega, \] where \(f\) is \(C^1\) and \(\nu\) denotes the exterior normal unit vector to \(\partial\Omega\), then \(\Omega\) is a ball, and \(u\) is radially symmetric and decreasing in the radial direction. This paper is concerned with a corresponding problem on an exterior domain. It is proved that if \(\Omega\) is an exterior domain with \(C^2\) boundary and \(u\in C^2(\overline\Omega)\) is a solution of \[ \Delta u+ f(u,|\nabla u|)= 0\quad\text{and} \quad 0\leq u<a\text{ in }\Omega, \]
\[ u= a\quad\text{and} \quad {\partial u\over\partial\nu}=\text{constant}\quad\text{on }\partial\Omega,\quad u,\;\nabla u= 0\text{ at }\infty, \] then \(\mathbb{R}^N\backslash\Omega\) is a ball, and \(u\) is radially symmetric and decreasing in the radial direction under some condition on the nonlinearity \(f\). The proof is done by the moving plane method due to A. D. Alexandroff and J. Serrin. As an application, the author proves a conjecture from potential theory due to P. Gruber. Precisely, it is proved that the only bounded \(C^{2,\alpha}\) domain \(D\) in \(\mathbb{R}^N\) \((N\geq 3)\) that admits a nontrivial single-layer potential which is constant in \(\overline D\) and is induced by a constant source distribution on \(\partial D\), is a ball.

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles in context of PDEs

Citations:

Zbl 0222.31007
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