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Serrin-type overdetermined problems: An alternative proof. (English) Zbl 1161.35025

Summary: We prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations, namely the Hessian equations. In the case of the Poisson equation, our proof is alternative to the proofs proposed by Serrin (moving planes) and by Weinberger. Moreover, our proof makes no direct use of the maximum principle while it sheds light on a relation between the Serrin problem and the isoperimetric inequality.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J60 Nonlinear elliptic equations
35R45 Partial differential inequalities and systems of partial differential inequalities
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