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Uniqueness results for nonlocal Hamilton–Jacobi equations

Barles, Guy; Cardaliaguet, Pierre; Ley, Olivier; Monteillet, Aurélien (2009), Uniqueness results for nonlocal Hamilton–Jacobi equations, Journal of Functional Analysis, 257, 5, p. 1261-1287. http://dx.doi.org/10.1016/j.jfa.2009.04.014

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00264757/fr/
Date
2009
Journal name
Journal of Functional Analysis
Volume
257
Number
5
Publisher
Elsevier
Pages
1261-1287
Publication identifier
http://dx.doi.org/10.1016/j.jfa.2009.04.014
Metadata
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Author(s)
Barles, Guy
Cardaliaguet, Pierre
Ley, Olivier
Monteillet, Aurélien
Abstract (EN)
We are interested in nonlocal eikonal equations describing the evolution of interfaces moving with a nonlocal, non-monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh–Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts.
Subjects / Keywords
Nonlocal Hamilton–Jacobi equations; Dislocation dynamics; Fitzhugh–Nagumo system; Nonlocal front propagation; Level-set approach; Geometrical properties; Lower-bound gradient estimate; Viscosity solutions; Eikonal equation; L1-dependence in time

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