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On Serrin’s overdetermined problem in space forms

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Abstract

We consider Serrin’s overdetermined problem for the equation \(\Delta v + nK v = -\,1\) in space forms, where K is the curvature of the space, and we prove a symmetry result by using a P-function approach. Our approach generalizes the one introduced by Weinberger to space forms and, as in the Euclidean case, it provides a short proof of the symmetry result which does not make use of the method of moving planes.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Giulio Ciraolo

  2. Dipartimento di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123, Torino, Italy

    Luigi Vezzoni

Authors
  1. Giulio Ciraolo
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  2. Luigi Vezzoni
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Corresponding author

Correspondence to Giulio Ciraolo.

Additional information

This work was partially supported by the project FIRB “Differential Geometry and Geometric functions theory” and FIR “Geometrical and Qualitative aspects of PDE”, and by GNSAGA and GNAMPA (INdAM) of Italy.

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Ciraolo, G., Vezzoni, L. On Serrin’s overdetermined problem in space forms. manuscripta math. 159, 445–452 (2019). https://doi.org/10.1007/s00229-018-1079-z

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  • DOI: https://doi.org/10.1007/s00229-018-1079-z

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