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Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations

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Abstract

We describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non-monotone second-order geometric equations arising in front propagation problems. Our method is based on some lower gradient bounds for the solution. These estimates are crucial to obtain regularity properties of the front, which allow to deal with nonlocal terms in the equations. Applications to short time uniqueness results for the initial value problems for dislocation type equations, asymptotic equations of a FitzHugh–Nagumo type system and equations depending on the Lebesgue measure of the fronts are presented.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et Physique Théorique, Fédération Denis Poisson, Université de Tours, Place de Grandmont, 37200, Tours, France

    Guy Barles

  2. IRMAR, INSA de Rennes, 35043, Rennes, France

    Olivier Ley

  3. Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

    Hiroyoshi Mitake

Authors
  1. Guy Barles
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  2. Olivier Ley
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  3. Hiroyoshi Mitake
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Correspondence to Olivier Ley.

Additional information

This work was partially supported by the ANR (Agence Nationale de la Recherche) through MICA project (ANR-06-BLAN-0082) and by the Research Fellowship (20-5332, 22-1725) for Young Researcher from JSPS and Excellent Young Researchers Overseas Visit Program of JSPS.

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Barles, G., Ley, O. & Mitake, H. Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations. Math. Ann. 352, 409–451 (2012). https://doi.org/10.1007/s00208-011-0648-1

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  • DOI: https://doi.org/10.1007/s00208-011-0648-1

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