Further PDE methods for weak KAM theory

Abstract

We introduce and make estimates for several new approximations that in appropriate asymptotic limits yield the key PDE for weak KAM theory, namely a HamiltonJacobi type equation for a potential u and a coupled transport equation for a measure σ.We revisit as well a singular variational approximation introduced in [E1], and demonstrate "approximate integrability" of certain phase space dynamics related to the Hamiltonian flow. Other examples include a pair of strongly coupled PDE suggested by the Lions-Lasry theory [L-L1] of mean field games and a new and extremely singular elliptic equation suggested by sup-norm variational theory.

Weak KAM theorem for Hamilton‐Jacobi equations with Neumann boundary conditions on noncompact manifolds

“…Since then, there have been a large amount of researches on Hamilton–Jacobi equation in view of weak KAM theory. For Hamilton–Jacobi equation confined in smooth compact manifolds, many valuable results had been obtained; see . Remarkably, in , the authors introduced a new kind of Lax‐Oleinik type operator with parameters and proved a family of new Lax‐Oleinik operators with an arbitrary continuous function as initial condition converges to a backward weak KAM solution of the corresponding Hamilton–Jacobi equation.…”

Elementary Symplectic Topology and Mechanics

“…Nevertheless, this new weak theoretical framework encouraged the construction of various approximate theories of integrability (see e.g. [20,23,58,59]). This last result goes back to [61,62,105] and others, an updated setup of this matter is presented by [40].…”

Convergence to the Time Average by Stochastic Regularization

“…We here provide such estimates with norms obtained by averaging over open domains of the phasespace. Therefore, our result has a probabilistic interpretation (probabilistic results in the weak KAM framework have been recently obtained by Evans, see [8], [9] and also [3]), whereas classical Hamiltonian Perturbation Theory provides uniform estimates valid for all initial conditions. The strength of the stochastic perturbation is a parameter of our construction.…”

“…global in the phase-space, inspired by standard viscosity and stochastic regularizations of PDEs (see, for example [11], [5]). We remark that, in the last years, a completely new approach to Hamiltonian dynamics motivated by regularization techniques has been represented by the so-called weak KAM theories (see [10], [14], [9]), which focus on the existence of the Aubry-Mather invariant sets.…”

“…The celebrated KAM and Nekhoroshev Theorems [12], [1], [16], [17] overcame this problem with a refined use of algebraic as well as geometric treatment of small divisors. More recently, the so-called weak KAM theories (see [10], [14], [9]) have studied the problem by new perspectives, based on variational and PDE regularizations by viscosity techniques.…”