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Mathematische Annalen

, Volume 352, Issue 2, pp 409–451 | Cite as

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

  • Guy Barles
  • Olivier LeyEmail author
  • Hiroyoshi Mitake
Article

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.

Mathematics Subject Classification (2000)

35K15 34A12 35A02 49L25 45K05 53C44 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Physique Théorique, Fédération Denis PoissonUniversité de ToursToursFrance
  2. 2.IRMAR, INSA de RennesRennesFrance
  3. 3.Department of Applied Mathematics, Graduate School of EngineeringHiroshima UniversityHigashi-HiroshimaJapan

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