A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative. (English) Zbl 1194.35282

Summary: We develop a new method to prove symmetry results for overdetermined boundary value problems. This method is based on the use of continuous Steiner symmetrization together with derivative with respect to the domain. It allows to consider nonlinear equations in divergence form with dependence in \(r=|x|\) in the nonlinearity. By using the notion of “local symmetry” introduced by the first author, we prove that the domain is necessarily a ball. We also give an example where the solution of the overdetermined problem is not radially symmetric.


35N10 Overdetermined systems of PDEs with variable coefficients
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