113 citations · 311 across the 24 of their papers we have counts for
26 papers
Selection of critical solutions in non-monotone vanishing discount problems
Panrui Ni, Jun Yan, Maxime Zavidovique
We study a generalized vanishing discount problem for Hamilton--Jacobi equations without assuming monotonicity either pointwise or in the averaged sense with respect to all Mather…
Integrability for conformally symplectic systems
Marie-Claude Arnaud, Xifeng Su, Maxime Zavidovique
The goal of this paper is to study the dynamics of conformally symplectic Hamiltonian flows under the light of integrability. As the dynamics of conformally symplectic Hamiltonian…
Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations
Andrea Davini, Panrui Ni, Jun Yan +1
We study the asymptotic behavior of solutions of an equation of the form \begin{equation}\label{abs}\tag{*} G\big(x, D_x u,λu(x)\big) = c_0\qquad\hbox{in } \end{equation} on a c…
Nonlinear and degenerate discounted approximation in discrete weak KAM theory
Panrui Ni, Maxime Zavidovique
In this paper, we introduce a discrete version of the nonlinear implicit Lax-Oleinik operator. We consider the associated vanishing discount problem with a non-degenerate condition…
Discrete and Continuous Weak KAM Theory: an introduction through examples and its applications to twist maps
Maxime Zavidovique
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 wea…
Convergence of the solutions of the nonlinear discounted Hamilton-Jacobi equation: The central role of Mather measures
Qinbo Chen, Albert Fathi, Maxime Zavidovique +1
Given a continuous Hamiltonian defined on , where is a closed connected manifold, we study viscosity solutions, $u_λ: M\…
Weak K.A.M. solutions and minimizing orbits of twist maps
Marie-Claude Arnaud, Maxime Zavidovique
For exact symplectic twist maps of the annulus, we etablish a choice of weak K.A.M. solutions that depend in a Lipschitz-continuous way on the cohomology class $c…
Actions of symplectic homeomorphisms/diffeomorphisms on foliations by curves in dimension 2
Marie-Claude Arnaud, Maxime Zavidovique
The two main results of this paper concern the regularity of the invariant foliation of a C0-integrable symplectic twist diffeomorphisms of the 2-dimensional annulus, namely that $…
Convergence of solutions for some degenerate discounted Hamilton--Jacobi equations
Maxime Zavidovique
We study solutions of Hamilton--Jacobi equations of the form where is a nonnegative function, a positive constant, a constant and…
Twisted Lax--Oleinik formulas and weakly coupled systems of Hamilton--Jacobi equations
Maxime Zavidovique
We show that viscosity solutions of evolutionary weakly coupled systems of Hamilton--Jacobi equations can be approximated by iterated twisted Lax--Oleinik like operators. We establ…
Fixed points of contractions approximating 1-Lipschitz maps
Maxime Zavidovique
A -Lipschitz map from a convex compact set to itself has fixed points. This consequence of Brouwer's or Schauder's fixed point theorem has more elementary proofs by approxim…
On the transversal dependence of weak K.A.M. solutions for symplectic twist maps
Marie-Claude Arnaud, Maxime Zavidovique
For a symplectic twist map, we prove that there is a choice of weak K.A.M. solutions that depend in a continuous way on the cohomology class. We thus obtain a continuous function $…
Convergence of the solutions of discounted Hamilton--Jacobi systems
Andrea Davini, Maxime Zavidovique
We consider a weakly coupled system of discounted Hamilton--Jacobi equations set on a closed Riemannian manifold. We prove that the corresponding solutions converge to a specific s…
Random Lax--Oleinik semigroups for Hamilton--Jacobi systems
Andrea Davini, Antonio Siconolfi, Maxime Zavidovique
Following the random approach of Mitake, Siconolfi,Tran and Yamada, we define a Lax--Oleinik formula adapted to evolutive weakly coupled systems of Hamilton--Jacobi equations. It i…
Convergence of the solutions of the discounted equation: the discrete case
Andrea Davini, Albert Fathi, Renato Iturriaga +1
We derive a discrete version of the results of our previous work. If is a compact metric space, a continuous cost function and , the u…
Convergence of the solutions of the discounted equation
Andrea Davini, Albert Fathi, Renato Iturriaga +1
We consider a continuous coercive Hamiltonian on the cotangent bundle of the compact connected manifold which is convex in the momentum. If is the visco…
On the (non) existence of viscosity solutions of multi-time Hamilton-Jacobi equations
Andrea Davini, Maxime Zavidovique
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.
Tonelli Hamiltonians without conjugate points and integrability
Marc Arcostanzo, Marie-Claude Arnaud, Philippe Bolle +1
We prove that all the Tonelli Hamiltonians defined on the cotangent bundle $T^*\T^n$ of the -dimensional torus that have no conjugate points are integrable, i.e. $T^*\T^n$…
Aubry sets for weakly coupled systems of Hamilton--Jacobi equations
Andrea Davini, Maxime Zavidovique
We introduce a notion of Aubry set for weakly coupled systems of Hamilton--Jacobi equations on the torus and characterize it as the region where the obstruction to the existence of…
Fast weak-KAM integrators for separable Hamiltonian systems
Anne Bouillard, Erwan Faou, Maxime Zavidovique
We consider a numerical scheme for Hamilton-Jacobi equations based on a direct discretization of the Lax-Oleinik semi-group. We prove that this method is convergent with respect to…
Regularization of subsolutions in discrete weak KAM theory
Patrick Bernard, Maxime Zavidovique
We expose different methods of regularizations of subsolutions in the context of discrete weak KAM theory. They allow to prove the existence and the density of subsolutio…
Weak KAM theoretic aspects for nonregular commuting Hamiltonians
Andrea Davini, Maxime Zavidovique
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 equiva…
Ilmanen's Lemma on Insertion of C Functions
Albert Fathi, Maxime Zavidovique
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.
Existence of critical subsolutions in discrete weak KAM theory
Maxime Zavidovique
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…