29 research outputs found
Existence of 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 defined on a smooth connected manifold
is locally semi-concave and verifies twist conditions, then there exists a
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 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 C 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 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
where is a nonnegative function, a positive constant,
a constant and a convex coercive Hamiltonian. Under suitable conditions on
we prove that the functions converge as to
a function that is a solution of the critical equation .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 , on a "reasonable" space . 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