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Gradient Bounds and Rigidity Results for Singular, Degenerate, Anisotropic Partial Differential Equations

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Abstract

We consider the Wulff-type energy functional,

$$\fancyscript{W}_\Omega(u) := \int_\Omega B( H( \nabla u (x) ) ) - F(u(x)) \, dx,$$

where B is positive, monotone and convex, and H is positive homogeneous of degree 1. The critical points of this functional satisfy a possibly singular or degenerate quasilinear equation in an anisotropic medium.

We prove that the gradient of the solution is bounded at any point by the potential F(u) and we deduce several rigidity and symmetry properties.

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Authors and Affiliations

  1. Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, UMR CNRS 7352, Université Picardie “Jules Verne”, 33 Rue St Leu, 80039, Amiens, France

    Matteo Cozzi & Alberto Farina

  2. Dipartimento di Matematica “Federigo Enriques”, Università degli studi di Milano, Via Saldini 50, 20133, Milano, Italy

    Matteo Cozzi & Enrico Valdinoci

  3. Institut “Camille Jordan”, UMR CNRS 5208, Université “Claude Bernard” Lyon I, 43 Boulevard du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Alberto Farina

  4. Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes”, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100, Pavia, Italy

    Enrico Valdinoci

  5. Weierstraß Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    Enrico Valdinoci

Authors
  1. Matteo Cozzi
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  2. Alberto Farina
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  3. Enrico Valdinoci
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Correspondence to Matteo Cozzi.

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Communicated by L.Caffarelli

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Cozzi, M., Farina, A. & Valdinoci, E. Gradient Bounds and Rigidity Results for Singular, Degenerate, Anisotropic Partial Differential Equations. Commun. Math. Phys. 331, 189–214 (2014). https://doi.org/10.1007/s00220-014-2107-9

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  • DOI: https://doi.org/10.1007/s00220-014-2107-9

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