Differentiability of Mather's beta function in low dimensions

TL;DR

This paper investigates the differentiability properties of Mather's beta function for time-periodic Lagrangians on a two-torus, establishing differentiability in multiple directions at various homology classes.

Contribution

It proves that Mather's beta function is differentiable in up to two directions at any k-irrational homology class on a two-torus.

Findings

Beta function is differentiable in at least k directions at k-irrational classes.

Differentiability holds for all k=0,1,2 on the two-torus.

Results extend understanding of the regularity of Mather's beta function.

Abstract

Let L be a time-periodic Lagrangian on a two-torus. Then the beta-function of L is differentiable at least in k directions at any k-irrational homology class, for k= 0, 1, 2.