Homogenization of viscous and non-viscous HJ equations: a remark and an application
Abstract
It was pointed out in [P.L. Lions, G. Papanicolaou, S. Varadhan, Homogenization of Hamilton-Jacobi equation, unpublished preprint (1987)] that, for first order Hamilton-Jacobi (HJ) equations, homogenization starting with affine initial data implies homogenization for general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 prove the above mentioned implication in both viscous and non-viscous cases. 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 and non-viscous HJ equations in one space dimension with non-convex Hamiltonians of specific form. The results are new in the viscous case.
Keywords
Cite
@article{arxiv.1608.01893,
title = {Homogenization of viscous and non-viscous HJ equations: a remark and an application},
author = {Andrea Davini and Elena Kosygina},
journal= {arXiv preprint arXiv:1608.01893},
year = {2016}
}
Comments
22 pages We have slightly changed the title and generalized the 1d homogenization results to a larger class of stationary ergodic Hamiltonians