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arXiv · 1608.01893

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

Andrea Davini·Elena Kosygina
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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It was pointed out in [P.L. Lions, G. Papanicolaou, S. Varadhan, Homogenization of Hamilton-Jacobi equation, unpublished preprint (1987)] that, for first order Hamilton-Jacobi (HJ) equations, homogenization starting with affine initial data implies homogenization for general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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Andrea Davini, Elena Kosygina. 2016-08-05. Homogenization of viscous and non-viscous HJ equations: a remark and an application. https://arxiv.org/abs/1608.01893

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