Homogenization of viscous and non-viscous HJ equations: a remark and an application

2016/08/05 by Davini, Andrea, Kosygina, Elena
#35B27 #35D40 #60K37 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1608.01893

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It was pointed out in [P.L. Lions, G. Papanicolaou, S. Varadhan, Homogenization of Hamilton-Jacobi equation, unpublished preprint (1987)] that, for first order Hamilton-Jacobi (HJ) equations, homogenization starting with affine initial data implies homogenization for general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.

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