Weak KAM Theorem for a Class of Infinite-Dimensional Lagrangian systems

TL;DR

This paper extends the Weak KAM theorem to a class of infinite-dimensional Lagrangian systems with specific potential function properties, establishing the existence of calibrated curves for each element in L^2(I).

Contribution

It introduces a novel application of the Weak KAM theorem to infinite-dimensional systems with periodic, rearrangement invariant, and weakly upper semicontinuous potentials.

Findings

Existence of calibrated curves for all M in L^2(I)

Extension of Weak KAM theorem to infinite-dimensional systems

Potential functions with specific invariance and semicontinuity properties

Abstract

In this paper, we study infinite-dimensional Lagrangian systems where the potential functions are periodic, rearrangement invariant and weakly upper semicontinuous. And we prove that there exists a calibrated curve for every $M\in L^{2}(I)$.