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Diversifications of Serrin’s and related symmetry problems. (English) Zbl 1269.35021

Summary: If \(D\) is a bounded \(C^1\) domain (in \(\mathbb{R}^n)\) for which the solution to the Dirichlet problem \[ \Delta u=-1,\quad\text{in }D,\quad u=0\quad\text{on }\partial D \] has the property that, for given constants \(r\), \(l>0\), and for all \(x\in\partial D\) \[ \text{dist}(x,\Gamma_l)=r, \quad(\Gamma_t=\{u=l\}), \] then \(D\) is necessarily a ball.
We prove this, and several other related symmetry results, using various known symmetry methods. The novelty of this article lies in the problem(s) rather than in the method(s). We also present (and in some cases also prove) a variety of possible formulations, that diversifies and generalizes Serrin’s and other symmetry problems.

MSC:

35N25 Overdetermined boundary value problems for PDEs and systems of PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
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