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

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 C2 in space and C1 in time. In order to do so, we combine viscosity solution techniques and Greenʹs function techniques. Viscosity solution theory provides the existence of a W1,∞ solution as well as uniqueness and stability results. A Duhamelʹs integral representation of the equation involving the Greenʹs function permits to prove further regularity. We also state the existence of C∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L∞ 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.