Homogenization of the Hamilton-jacobi Equation

Since the celebrated work by Lions, Papanicolaou and Varadhan in 1980’s, there has been a considerable attention to the homogenization problem for Hamilton-Jacobi equation: roughly speaking, how to describe what macroscopic properties and aspects of this equation survive, once all of its local features are neglected (for example, by averaging over faster and faster oscillations). The interest in this question – besides the importance of the Hamilton-Jacobi equation in many different contexts (classical mechanics, symplectic geometry, PDEs, etc...) – has been in recent years boosted by the manifold connections that it shares with new prominent areas of research: Aubry-Mather theory, weak KAM theory, symplectic homogenization, just to mention a few of them. In this article, we discuss a very natural and important question, namely how to extend these classical results beyond the Euclidean setting, which has been recently addressed in a work by Contreras, Iturriaga and Siconolfi. Starting from their result and from the above-mentioned work by Lion, Papanicolaou and Varadhan, we first describe how both of them can be interpreted as a particular case of a more intrinsic and geometric approach, i.e., the case of Tonelli Hamiltonians which are invariant under the action of a discrete group. In particular, we point out the leading rôle played by the algebraic nature of the group (more specifically, its rate of growth) in driving the homogenization process and determining the features of the limit problem. Then, we prove a homogenization result in the case of Hamiltonians that are invariant under the action of a discrete (virtually) nilpotent group (i.e., with polynomial growth). Besides being more intrinsic and geometric, this novel approach provides a much clearer understanding of the structures of both the limit space and the homogenized equation, unveiling features and phenomena that in the previously-studied cases were shadowed by either the homogeneity of the ambient space or the abelianity of the acting group.