Homogenization of viscous Hamilton-Jacobi equations: a remark and an application
Andrea Davini, Elena Kosygina · arXiv (Cornell University) · 2016
It was pointed out in [P.L. Lions, G. Papanicolaou, S. Varadhan, Homogenization of Hamilton-Jacobi equation, unpublished preprint (1987)] that, for first order Hamilton-Jacobi (HJ) equations, homogenization starting with affine initial data implies homogenization for general uniformly continuous initial data. The argument makes use of some properties of the HJ semi-group, in particular, the finite speed of propagation. The last property is lost for viscous HJ equations. In this paper, we give a proof of this fact both in the viscous and in the non-viscous case. Our proof relies on a variant of Evans's perturbed test function method. As an application, we show homogenization in the stationary ergodic setting for viscous HJ equations in one space dimension with non-convex Hamiltonians of specific form.