A Weak KAM theorem; from finite to infinite dimension

These notes contain a series of lectures given by the first author in the 2008 GNAMPA– INDAM School in Pisa. It is based on recent results by both authors who initiated a study of an infinite dimensional weak KAM theory. While some of the results presented here have already appeared in their joint work [16], the core of this manuscript, section 3, has never been submitted for publication anywhere. The current manuscript should be regarded as a companion to [16]. In [16] it is shown that asymptotic behaviour of a class of Vlasov systems can be studied via a cell problem (C): H(M, c+∇U) = H̄(c). (C) is to be satisfied in the sense of viscosity on what will be referred to as the L(0, 1)–torus (cf. (10)), a quotient space of the Hilbert space L(0, 1). More importantly, existence of solutions U for (C) and existence of calibrated curves associated to U are obtained by studying the limit as ǫ tend to zero of Hamilton-Jacobi equations of the form (HJE)ǫ: ǫV + H(M, c +∇V ) = 0. The purpose of these lectures is to show that if H satisfies appropriate invariance properties, a Galerkin type approximation method can be used to establish existence of solutions for (HJE)ǫ. This result is in contrast with [8], where it is shown that Galerkin type methods are not expected to provide solutions for (HJE)ǫ. We could have identified the largest class of Hamiltonians for which the results in these notes hold. We chose not to work in the greatest generality for two reasons: first of all, our study was motivated by the Vlasov systems appearing in kinetic theory and we restrict ourselves to Hamiltonians corresponding to these PDEs. Secondly, we tried to keep the computations as simple as possible to separate the main ideas from technical details.