No abstract

We propose a PDE approach to the Aubry-Mather theory using viscosity solutions. This allows to treat Hamiltonians (on the flat torus T-N) just coercive. continuous and quasiconvex, for which a Hamiltonian flow cannot necessarily be defined. The analysis is focused on the family of Hamilton-Jacobi equations H(x, Du) = a with a real parameter, and in particular on the unique equation of the family. corresponding to the so-called critical value a = c, for which there is a viscosity solution on T-N. We define generalized projected Aubry and Mather sets and recover several properties of these sets holding for regular Hamiltonians

We consider the Hamilton-Jacobi equation ∂tu + H(x, Du) = 0 in (0, +∞) × T N where T N is the flat N-dimensional torus, and the Hamiltonian H(x, p) is assumed continuous in x and strictly convex and coercive in p. We study the large time behavior of solutions, and we identify the limit through a Lax-type formula. Some convergence results are also given for H solely convex. Our qualitative method is based on the analysis of the dynamical properties of the so called Aubry set, performed in the spirit of [13]. This can be viewed as a generalization of the techniques used in [11] and [18]. Analogous results have been obtained in [4] using PDE methods.

We are concerned with the existence of smooth time functions on connected time-oriented Lorentzian manifolds. The problem is tackled in a more general abstract setting, namely in a manifold M where is just defined a field of tangent convex cones (C-x)(x is an element of M) enjoying mild continuity properties. Under some conditions on its integral curves, we will construct a time function. Our approach is based on the definition of an intrinsic length for curves indicating how a curve is far from being an integral trajectory of C-x. We find connections with topics pertaining to Hamilton-Jacobi equations, and make use of tools and results issued from weak KAM theory

We perform a qualitative investigation of critical Hamilton-Jacobi equations, with stationary ergodic Hamiltonian, in dimension 1. We show the existence of approximate correctors, give characterizing conditions for the existence of correctors, provide Lax-type representation formulae and establish comparison principles. The results are applied to look into the corresponding effective Hamiltonian and to study a homogenization problem. In the analysis a crucial role is played by tools from stochastic geometry such as, for instance, closed random stationary sets