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.
Brandolini B., Nitsch C., Salani P., Trombetti C. (2008). Serrin-type overdetermined problems: An alternative proof. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 190(2), 267-280 [10.1007/s00205-008-0119-3].
Serrin-type overdetermined problems: An alternative proof
Brandolini B.
;
2008-01-01
Abstract
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.
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