Existence of C1,1C^{1,1} critical subsolutions in discrete weak KAM theory

In this article, following a first work of the author, we study critical subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function $c:M \times M\to \R{}$ defined on a smooth connected manifold is locally semi-concave and verifies twist conditions, then there exists a $C^{1,1}$ critical subsolution strict on a maximal set (namely, outside of the Aubry set). We also explain how this applies to costs coming from Tonelli Lagrangians. Finally, following ideas introduced in the work of Fathi-Maderna and Mather, we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analogue of Mather's $\alpha$ function on the cohomology.Comment: 28 page

Discrete and Continuous Weak KAM Theory: an introduction through examples and its applications to twist maps

The aim of these notes is to present a self contained account of discrete weak KAM theory. Put aside the intrinsic elegance of this theory, it is also a toy model for classical weak KAM theory, where many technical difficulties disappear, but where central ideas and results persist. It can therefore serve as a good introduction to (continuous) weak KAM theory. After a general exposition of the general abstract theory, several examples are studied. The last section is devoted to the historical problem of conservative twist maps of the annulus. At the end of the first three Chapters, the relations between the results proved in the discrete setting and the analogous theorems of classical weak KAM theory are discussed. Some key differences are also highlighted between the discrete and classical theory. Those results are new. The text also contains other results never published before, such as the convergence of solutions of discounted equations for degenerate perturbations.Comment: 166 pages, 7 figures. Extracted from the author's habilitation (HDR). Intended to be published as a pedagogical boo

On the (non) existence of viscosity solutions of multi--time Hamilton--Jacobi equations

We prove that the multi--time Hamilton--Jacobi equation in general cannot be solved in the viscosity sense, in the non-convex setting, even when the Hamiltonians are in involution.Comment: 15 page

Ilmanen's Lemma on Insertion of C1,1^{1,1} Functions

We give a proof of Ilmanen's lemma, which asserts that between a locally semi-convex and a locally semi-concave function it is possible to find a C$^{1,1}$ function.Comment: 17 pages, 1 figure, accepted for publication in Rend. Semin. Mat. Univ. Padov

Convergence of solutions for some degenerate discounted Hamilton--Jacobi equations

We study solutions of Hamilton--Jacobi equations of the form $$\lambda \alpha(x) u_\lambda(x) + H(x, D_x u_\lambda) = c,$$ where $\alpha$ is a nonnegative function, $\lambda$ a positive constant, $c$ a constant and $H$ a convex coercive Hamiltonian. Under suitable conditions on $\alpha$ we prove that the functions $u_\lambda$ converge as $\lambda\to 0$ to a function $u_0$ that is a solution of the critical equation $H(x, D_x u_0) = c$.Comment: Version 2 ; 25 pages, modified according to the comments and suggestions by referees to improve presentation. Accepted for publication in Analysis & PD

Strict sub-solutions and Ma\~ne potential in discrete weak KAM theory

In this paper, we explain some facts on the discrete case of weak KAM theory. In that setting, the Lagrangian is replaced by a cost $c:X\times X \to \mathbb{R}$, on a "reasonable" space $X$. This covers for example the case of periodic time-dependent Lagrangians. As is well known, it is possible in that case to adapt most of weak KAM theory. A major difference is that critical sub-solutions are not necessarily continuous. We will show how to define a Ma\~ne potential. In contrast to the Lagrangian case, this potential is not continuous. We will recover the Aubry set from the set of continuity points of the Ma\~ne potential, and also from critical sub-solutions.Comment: 49 page

Weak KAM theoretic aspects for nonregular commuting Hamiltonians

In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax- Oleinik semigroups. This is equivalent to the solvability of an associated multi- time Hamilton-Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions and the same Aubry set, thus generalizing a result recently obtained by the second author for Tonelli Hamiltonians. We make a further step by proving that the Hamiltonians admit a common critical subsolution, strict outside their Aubry set. This subsolution can be taken of class C^{1,1} in the Tonelli case. To prove our main results in full generality, it is crucial to establish suitable differentiability properties of the critical subsolutions on the Aubry set. These latter results are new in the purely continuous case and of independent interest.Comment: 37 pages. Third version. Presentation of the commutation property changed. Proof of the main theorem made cleare