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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https://hal.archives-ouvertes.fr/hal-00176542
Contributor : Cyril Imbert <>
Submitted on : Wednesday, October 3, 2007 - 11:04:28 PM
Last modification on : Thursday, January 11, 2018 - 6:15:40 AM
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  • HAL Id : hal-00176542, version 1

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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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