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Serrin’s result for domains with a corner or cusp. (English) Zbl 0943.35022

The author proves that, if \(u\) is a positive solution of the differential equation \(\Delta u+1= 0\) on a bounded domain \(\Omega\subset \mathbb{R}^n\) with boundary conditions \[ u= 0,\quad {\partial u\over\partial\nu}= c\quad\text{on }\partial\Omega\setminus \{P\} \] then \(\Omega\) is a ball and \(u\) is radially symmetric.
Here \(c\) is a constant, \(P\in \partial\Omega\), and \({\partial u\over\partial\nu}\) is the normal derivative along the inward normal field on \(\partial\Omega\). Here, the normal derivative is defined everywhere except at a possible corner or a cusp at \(P\). The boundary of \(\Omega\) satisfies either the interior sphere condition or the exterior sphere condition.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B99 Qualitative properties of solutions to partial differential equations
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References:

[1] J. Serrin, A symmetry problem in potential theory , Arch. Rational Mech. Anal. 43 (1971), 304-318. · Zbl 0222.31007
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