In the context of Mathers theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian.
We consider the variational problem consisting of minimizing a polyconvex integrand for maps between manifolds. We offer a simple and direct proof of the existence of a minimizing map. The proof is based on Young measures.
Nonnegative measure-valued solutions of the continuity equation are uniquely determined by their initial condition, if the characteristic ODE associated to the velocity field has a unique solution. In this paper we give a partial extension of this re
sult to signed measure-valued solutions, under a quantitative two-sided Osgood condition on the velocity field. Our results extend those obtained for log-Lipschitz vector fields by Bahouri and Chemin.
