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Radial symmetry of overdetermined boundary-value problems in exterior domains. (English) Zbl 0911.35008

Summary: We extend a classical result by J. Serrin to exterior domains \(\mathbb{R}^n\setminus\overline\Omega\), where \(\Omega\) is a bounded domain. We prove, under some hypotheses on \(f\), that if there exists a solution of \(\Delta u+f(u)=0\) in \(\mathbb{R}^n\setminus\overline\Omega\) satisfying the overdetermined boundary conditions that \(\partial u/\partial v\) and \(u\) are constant on \(\partial\Omega\), and such that \(0\leq u\leq u|_{\partial\Omega}\), then the domain \(\Omega\) is a ball. Under different assumptions on \(f\), this result has been obtained by W. Reichel. The main result here covers new cases like \(f(u)=u^p\) with \(n/(n-2)<p\leq(n+2)/(n-2)\). When \(\Omega\) is a ball, almost the same proof allows us to derive the symmetry of positive bounded solutions satisfying only the Dirichlet condition that \(u\) is constant on \(\partial\Omega\). Our method relies on Kelvin transforms, various forms of the maximum principle and the device of moving planes up to a critical position.

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35N05 Overdetermined systems of PDEs with constant coefficients
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