# A non-local regularization of first order Hamilton-Jacobi equations

Abstract : In this paper, we investigate the regularizing effect of a non-local operator on first order Hamilton-Jacobi equations. We prove that there exists a unique solution that is $C^2$ in space and $C^1$ in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a $W^{1,\infty}$ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of $C^\infty$ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in $L^\infty$ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part.
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Article dans une revue
Journal of Differential Equations, Elsevier, 2005, 211 (1), pp.218-246

https://hal.archives-ouvertes.fr/hal-00176542
Contributeur : Cyril Imbert <>
Soumis le : mercredi 3 octobre 2007 - 23:04:28
Dernière modification le : jeudi 11 janvier 2018 - 06:15:40
Document(s) archivé(s) le : jeudi 27 septembre 2012 - 12:48:02

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• HAL Id : hal-00176542, version 1

### Citation

Cyril Imbert. A non-local regularization of first order Hamilton-Jacobi equations. Journal of Differential Equations, Elsevier, 2005, 211 (1), pp.218-246. 〈hal-00176542〉

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