16 feb 2005

Abstract.

Seminari di calcolo delle variazioni

when: Wednesday, February 16, at 6 pm

where: Departiment of Mathematics, Sala delle Riunioni

speaker: Antonio Siconolfi (Roma I)

title: Aubry set and applications

abstract: For a given Hamiltonian $H(x,p)$ continuous and quasiconvex in the second argument, defined in $R^N \times R^N$ or on the cotangent bundle of a compact boundaryless manifold, we consider the equation $H=c$ with $c$  critical value,  i.e. for which the equation admits locally Lipschitz--continuous a.e. subsolutions, but not strict subsolutions. We show that there is a subset of the state variable space, called Aubry set and denoted by  $\A$, where the obstruction to the existence of such subsolution is concentrated. We give a metric characterization of $\A$, and we discuss its main properties. They are applied to a projection problem in a Banach space, to the study of the large-time behavior of subsolutions to a time-dependent Hamilton-Jacobi equation, and to construct a Lyapunov function for a perturbed dynamics, under suitable stability assumptions.