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Sobolev regularity for the first order Hamilton–Jacobi equation

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Abstract

We provide Sobolev estimates for solutions of first order Hamilton–Jacobi equations with Hamiltonians which are superlinear in the gradient variable. We also show that the solutions are differentiable almost everywhere. The proof relies on an inverse Hölder inequality. Applications to mean field games are discussed.

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Acknowledgments

The authors wish to thank the anonymous referee for the careful reading of the paper and the suggestions. Supported by the Italian Indam Gnampa project 2013 “Modelli di campo medio nelle dinamiche di popolazioni e giochi differenziali”. The first author wishes to thank the Indam and the University Rome Tor Vergata where the project was partially completed.

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Authors and Affiliations

  1. Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Pierre Cardaliaguet & Daniela Tonon

  2. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133, Roma, Italy

    Alessio Porretta

Authors
  1. Pierre Cardaliaguet
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  2. Alessio Porretta
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  3. Daniela Tonon
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Corresponding author

Correspondence to Pierre Cardaliaguet.

Additional information

Communicated by L. Ambrosio.

Cardaliaguet was partially supported by the ANR (Agence Nationale de la Recherche) projects ANR-10-BLAN 0112, ANR-12-BS01-0008-01 and ANR-14-ACHN-0030-01.

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Cardaliaguet, P., Porretta, A. & Tonon, D. Sobolev regularity for the first order Hamilton–Jacobi equation. Calc. Var. 54, 3037–3065 (2015). https://doi.org/10.1007/s00526-015-0893-3

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  • DOI: https://doi.org/10.1007/s00526-015-0893-3

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