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Journal of Differential Equations 211, 1 (2005) 218-246

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

Cyril Imbert 1

(2005)

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.

1:	Institut de Mathématiques et de Modélisation de Montpellier (I3M)
	CNRS : UMR5149 – Université Montpellier II - Sciences et Techniques du Languedoc

Subject

:

Mathematics/Analysis of PDEs

Keyword(s): integro-differential Hamilton-Jacobi equation – non-local regularization – Lévy operator – viscosity solution – error estimate

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From: Cyril Imbert <>

Submitted on: Wednesday, 3 October 2007 23:04:28

Updated on: Thursday, 4 October 2007 09:07:18