Calculus of Variations and Geometric Measure Theory

A. Figalli - L. Rifford

Closing Aubry sets I

created by figalli on 19 Jul 2013


Accepted Paper

Inserted: 19 jul 2013
Last Updated: 19 jul 2013

Journal: Comm. Pure Appl. Math.
Year: 2013


Given a Tonelli Hamiltonian $H:T^*M \rightarrow \R$ of class $C^k$, with $k\geq 2$, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set $\bar x$, and a critical viscosity subsolution $u$, such that $u$ is a $C^{1}$ critical solution in an open neighborhood of the positive orbit of $\bar x$. Suppose further that $u$ is ``$C^2$ at $\bar x$''. Then there exists a $C^k$ potential $V:M \rightarrow \R$, small in $C^2$ topology, for which the Aubry set of the new Hamiltonian $H+V$ is either an equilibrium point or a periodic orbit. (2) If $M$ is two dimensional, (1) holds replacing ``$C^{1}$ critical solution + $C^2$ at $\bar x$'' by ``$C^3$ critical subsolution''.

These results can be considered as a first step through the attempt of proving the Ma\~n\'e's conjecture in $C^2$ topology. In a second paper we will generalize (2) to arbitrary dimension. Moreover, such an extension will need the introduction of some new techniques, which will allow us to prove the Ma\~n\'e's density Conjecture in $C^1$ topology. Our proofs are based on a combination of techniques coming from finite dimensional control theory and Hamilton-Jacobi theory, together with some of the ideas which were used to prove $C^1$-closing lemmas for dynamical systems.