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Serrin-Type Overdetermined Problems: an Alternative Proof

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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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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso Monte S. Angelo, via Cintia, 80126, Naples, Italy

    B. Brandolini, C. Nitsch & C. Trombetti

  2. Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134, Florence, Italy

    P. Salani

Authors
  1. B. Brandolini
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  2. C. Nitsch
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  3. P. Salani
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  4. C. Trombetti
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Correspondence to C. Trombetti.

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Communicated by L. Ambrosio

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Brandolini, B., Nitsch, C., Salani, P. et al. Serrin-Type Overdetermined Problems: an Alternative Proof. Arch Rational Mech Anal 190, 267–280 (2008). https://doi.org/10.1007/s00205-008-0119-3

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  • DOI: https://doi.org/10.1007/s00205-008-0119-3

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