Chinese Annals of Mathematics, Series B, ISSN 0252-9599, 3/2017, Volume 38, Issue 2, pp. 379 - 392

This paper records for the Hamiltonian H = 1/2 |p|2 + W(x) some old and new identities relevant for the PDE/variational approach to weak KAM theory.

35A15 | Effective Hamiltonian | Mathematics, general | Weak KAM theory | Mathematics | 37J40 | Applications of Mathematics | Hamiltonian dynamics | MATHEMATICS | PDE METHODS | LAGRANGIAN SYSTEMS

35A15 | Effective Hamiltonian | Mathematics, general | Weak KAM theory | Mathematics | 37J40 | Applications of Mathematics | Hamiltonian dynamics | MATHEMATICS | PDE METHODS | LAGRANGIAN SYSTEMS

Journal Article

Discrete and Continuous Dynamical Systems- Series A, ISSN 1078-0947, 11/2016, Volume 36, Issue 11, pp. 6167 - 6185

We develop several aspects of the in finite-dimensional Weak KAM theory using a random variables' approach...

Hamilton-Jacobi equations | Weak KAM theory | Voscosity solutions | Dynamical systems | LAX-OLEINIK SEMIGROUP | EXISTENCE | VISCOSITY SOLUTIONS | HILBERT-SPACES | MATHEMATICS, APPLIED | weak KAM theory | MATHEMATICS | VLASOV EQUATION | VERSION | DYNAMICS | SYSTEMS | COMPACT MANIFOLDS | voscosity solutions

Hamilton-Jacobi equations | Weak KAM theory | Voscosity solutions | Dynamical systems | LAX-OLEINIK SEMIGROUP | EXISTENCE | VISCOSITY SOLUTIONS | HILBERT-SPACES | MATHEMATICS, APPLIED | weak KAM theory | MATHEMATICS | VLASOV EQUATION | VERSION | DYNAMICS | SYSTEMS | COMPACT MANIFOLDS | voscosity solutions

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 2019, Volume 267, Issue 4, pp. 2448 - 2470

For mechanical Hamiltonian systems on the torus, we study the dynamical properties of the generalized characteristic semiflows associated with the...

Generalized characteristic | Weak KAM theory | Hamilton-Jacobi equation | Singularities | TOTAL DISCONNECTEDNESS | MATHEMATICS | VISCOSITY SOLUTIONS | SET | HOMOCLINIC ORBITS | LASRY-LIONS

Generalized characteristic | Weak KAM theory | Hamilton-Jacobi equation | Singularities | TOTAL DISCONNECTEDNESS | MATHEMATICS | VISCOSITY SOLUTIONS | SET | HOMOCLINIC ORBITS | LASRY-LIONS

Journal Article

Discrete and Continuous Dynamical Systems - Series B, ISSN 1531-3492, 01/2013, Volume 18, Issue 1, pp. 57 - 94

.... 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...

Multi-time Hamilton-Jacobi equation | Commuting Hamiltonians | Weak KAM theory | Aubry-Mather theory | Viscosity solutions | multi-time Hamilton-Jacobi equation | MATHEMATICS, APPLIED | weak KAM theory | JACOBI EQUATIONS | MATHER THEORY | CONVEX HAMILTONIANS | MULTITIME EQUATIONS | CONTINUITY | REGULARITY | LARGE TIME BEHAVIOR | SYSTEMS | MANIFOLDS | viscosity solutions

Multi-time Hamilton-Jacobi equation | Commuting Hamiltonians | Weak KAM theory | Aubry-Mather theory | Viscosity solutions | multi-time Hamilton-Jacobi equation | MATHEMATICS, APPLIED | weak KAM theory | JACOBI EQUATIONS | MATHER THEORY | CONVEX HAMILTONIANS | MULTITIME EQUATIONS | CONTINUITY | REGULARITY | LARGE TIME BEHAVIOR | SYSTEMS | MANIFOLDS | viscosity solutions

Journal Article

Minimax Theory and its Applications, ISSN 2199-1413, 2018, Volume 3, Issue 2, pp. 313 - 322

Journal Article

Dynamic Games and Applications, ISSN 2153-0785, 12/2013, Volume 3, Issue 4, pp. 473 - 488

.... The well-posedness of the latter system and the uniqueness of the ergodic constant rely on weak KAM theory.

Computer Systems Organization and Communication Networks | Economics/Management Science, general | Mean field game | Communications Engineering, Networks | Weak KAM theory | Mathematics | Operations Research, Management Science | Game Theory/Mathematical Methods | Long time average | Game Theory, Economics, Social and Behav. Sciences | HAMILTON-JACOBI EQUATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | BEHAVIOR | Optimization and Control

Computer Systems Organization and Communication Networks | Economics/Management Science, general | Mean field game | Communications Engineering, Networks | Weak KAM theory | Mathematics | Operations Research, Management Science | Game Theory/Mathematical Methods | Long time average | Game Theory, Economics, Social and Behav. Sciences | HAMILTON-JACOBI EQUATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | BEHAVIOR | Optimization and Control

Journal Article

Commentarii Mathematici Helvetici, ISSN 0010-2571, 2012, Volume 87, Issue 1, pp. 1 - 39

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 $c\colon X\times X \rightarrow \mathbb{R}$, on a “reasonable” space $X...

Partial differential equations | General | Dynamical systems and ergodic theory | MATHEMATICS | SUBSOLUTIONS | MANIFOLDS | Discrete weak KAM theory | Aubry-Mather theory | continuous and discontinuous critical sub-solutions

Partial differential equations | General | Dynamical systems and ergodic theory | MATHEMATICS | SUBSOLUTIONS | MANIFOLDS | Discrete weak KAM theory | Aubry-Mather theory | continuous and discontinuous critical sub-solutions

Journal Article

Communications in Contemporary Mathematics, ISSN 0219-1997, 04/2013, Volume 15, Issue 2, pp. 1250055 - 1250036

We consider a recent approximate variational principle for weak KAM theory proposed by Evans...

Weak KAM theory | Hamilton-Jacobi equation | perturbation theory | approximate variational principles | MATHEMATICS | MATHEMATICS, APPLIED | PDE METHODS | LAGRANGIAN SYSTEMS | ORBITS | Lower bounds | Degrees of freedom | Approximation | Dynamics | Mathematical analysis | Variational principles | Estimates | Dynamical systems

Weak KAM theory | Hamilton-Jacobi equation | perturbation theory | approximate variational principles | MATHEMATICS | MATHEMATICS, APPLIED | PDE METHODS | LAGRANGIAN SYSTEMS | ORBITS | Lower bounds | Degrees of freedom | Approximation | Dynamics | Mathematical analysis | Variational principles | Estimates | Dynamical systems

Journal Article

Commentarii Mathematici Helvetici, ISSN 0010-2571, 2012, Volume 87, Issue 1, pp. 1 - 39

Journal Article

10.
Full Text
Weak KAM aspects of convex Hamilton–Jacobi equations with Neumann type boundary conditions

Journal de mathématiques pures et appliquées, ISSN 0021-7824, 2011, Volume 95, Issue 1, pp. 99 - 135

... of R n with the Neumann type boundary condition D γ u = g in the viewpoint of weak KAM theory, where γ...

Aubry–Mather theory | Weak KAM theory | Neumann type boundary conditions | Hamilton–Jacobi equations | Viscosity solutions | Hamilton-Jacobi equations | Aubry-Mather theory | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | LARGE-TIME BEHAVIOR | ASYMPTOTIC SOLUTIONS | MATHEMATICS | PARTIAL-DIFFERENTIAL-EQUATIONS | CONVERGENCE | OBLIQUE DERIVATIVE PROBLEMS

Aubry–Mather theory | Weak KAM theory | Neumann type boundary conditions | Hamilton–Jacobi equations | Viscosity solutions | Hamilton-Jacobi equations | Aubry-Mather theory | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | LARGE-TIME BEHAVIOR | ASYMPTOTIC SOLUTIONS | MATHEMATICS | PARTIAL-DIFFERENTIAL-EQUATIONS | CONVERGENCE | OBLIQUE DERIVATIVE PROBLEMS

Journal Article

Advanced Nonlinear Studies, ISSN 1536-1365, 2013, Volume 13, Issue 4, pp. 853 - 866

This paper contributes several results on weak KAM theory for time-periodic Tonelli Lagrangian systems. Wang and Yan [Commun. Math. Phys. 309 (2012), 663-691...

New Lax-Oleinik type operators | Hamilton-Jacobi equations | Weak KAM theory | Time-periodic Tonelli Lagrangians | LAX-OLEINIK SEMIGROUP | MATHEMATICS | MATHEMATICS, APPLIED | time-periodic Tonelli Lagrangians | PSEUDOGRAPHS | ORBITS | CONVERGENCE | weak KAM theory | new Lax-Oleinik type operators

New Lax-Oleinik type operators | Hamilton-Jacobi equations | Weak KAM theory | Time-periodic Tonelli Lagrangians | LAX-OLEINIK SEMIGROUP | MATHEMATICS | MATHEMATICS, APPLIED | time-periodic Tonelli Lagrangians | PSEUDOGRAPHS | ORBITS | CONVERGENCE | weak KAM theory | new Lax-Oleinik type operators

Journal Article

Mathematische Zeitschrift, ISSN 0025-5874, 06/2015, Volume 280, Issue 1-2, pp. 165 - 194

Journal Article

Dynamic games and applications, ISSN 2153-0793, 2019, Volume 10, Issue 2, pp. 361 - 390

The aim of this paper is to study the long-time behavior of solutions to deterministic mean field games systems on Euclidean space. This problem was addressed...

Weak KAM theory | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | WEAK KAM THEOREM | Mean field games | Long-time behavior | Viscosity solutions | Mathematics - Optimization and Control

Weak KAM theory | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | WEAK KAM THEOREM | Mean field games | Long-time behavior | Viscosity solutions | Mathematics - Optimization and Control

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 09/2020, Volume 269, Issue 7, pp. 5730 - 5753

We investigated several global behaviors of the weak KAM solutions uc(x,t) parametrized by c∈H1(T,R). For the suspended Hamiltonian H...

Exact syplectic twist map | Aubry Mather theory | Hamilton Jacobi equation | Generalized characteristics | Transition chain | Weak KAM solution

Exact syplectic twist map | Aubry Mather theory | Hamilton Jacobi equation | Generalized characteristics | Transition chain | Weak KAM solution

Journal Article

Discrete and Continuous Dynamical Systems- Series A, ISSN 1078-0947, 11/2016, Volume 36, Issue 11, pp. 6487 - 6522

.... By introducing an implicitly defined solution semigroup and an admissible value set C-H, we extend weak KAM theory to certain more general cases, in which H depends on the unknown function u explicitly...

Hamilton-Jacobi equations | Weak KAM theory | Viscosity solutions | LAX-OLEINIK SEMIGROUP | MATHEMATICS | MATHEMATICS, APPLIED | DEFINITE LAGRANGIAN SYSTEMS | LONG-TIME BEHAVIOR | CONVERGENCE | FRENKEL-KONTOROVA MODEL | viscosity solutions

Hamilton-Jacobi equations | Weak KAM theory | Viscosity solutions | LAX-OLEINIK SEMIGROUP | MATHEMATICS | MATHEMATICS, APPLIED | DEFINITE LAGRANGIAN SYSTEMS | LONG-TIME BEHAVIOR | CONVERGENCE | FRENKEL-KONTOROVA MODEL | viscosity solutions

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 01/2016, Volume 85, Issue 297, pp. 85 - 117

geometric integrator satisfying a discrete weak-KAM theorem which allows us to control its long time behavior...

Hamilton-Jacobi equations | min | Plus) convolution | Weak-KAM theorem | Geometric integration | WEIGHTED ENO SCHEMES | MATHEMATICS, APPLIED | THEOREM | LAGRANGIAN SYSTEMS | weak-KAM theorem | JACOBI EQUATIONS | CONVERGENCE | TRANSFORM | (min, plus) convolution | CONSERVATION-LAWS | APPROXIMATION SCHEMES | Numerical Analysis | Mathematics

Hamilton-Jacobi equations | min | Plus) convolution | Weak-KAM theorem | Geometric integration | WEIGHTED ENO SCHEMES | MATHEMATICS, APPLIED | THEOREM | LAGRANGIAN SYSTEMS | weak-KAM theorem | JACOBI EQUATIONS | CONVERGENCE | TRANSFORM | (min, plus) convolution | CONSERVATION-LAWS | APPROXIMATION SCHEMES | Numerical Analysis | Mathematics

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 2018, Volume 265, Issue 2, pp. 719 - 732

We study the Lasry–Lions approximation using the kernel determined by the fundamental solution with respect to a time-dependent Tonelli Lagrangian. This...

Aubry–Mather theory | Weak KAM theory | Hamilton–Jacobi equations | Lasry–Lions regularization | MATHEMATICS | Hamilton-Jacobi equations | HILBERT-SPACES | SET | LAX-OLEINIK | LEMMA | POINT-OF-VIEW | DYNAMICS | Lasry-Lions regularization | Aubry-Mather theory | REGULARIZATION

Aubry–Mather theory | Weak KAM theory | Hamilton–Jacobi equations | Lasry–Lions regularization | MATHEMATICS | Hamilton-Jacobi equations | HILBERT-SPACES | SET | LAX-OLEINIK | LEMMA | POINT-OF-VIEW | DYNAMICS | Lasry-Lions regularization | Aubry-Mather theory | REGULARIZATION

Journal Article

18.
Full Text
Existence and regularity of strict critical subsolutions in the stationary ergodic setting

Annales de l'Institut Henri Poincaré / Analyse non linéaire, ISSN 0294-1449, 03/2016, Volume 33, Issue 2, pp. 243 - 272

We prove that any continuous and convex stationary ergodic Hamiltonian admits critical subsolutions, which are strict outside the random Aubry set. They make...

Homogenization | Weak KAM Theory | Stationary ergodic setting | Viscosity solutions | WEAK KAM | MATHEMATICS, APPLIED | BOLZA PROBLEMS | HAMILTON-JACOBI EQUATIONS | MANIFOLDS | DISCONTINUOUS LAGRANGIANS

Homogenization | Weak KAM Theory | Stationary ergodic setting | Viscosity solutions | WEAK KAM | MATHEMATICS, APPLIED | BOLZA PROBLEMS | HAMILTON-JACOBI EQUATIONS | MANIFOLDS | DISCONTINUOUS LAGRANGIANS

Journal Article

Dynamic games and applications, ISSN 2153-0793, 2016, Volume 7, Issue 4, pp. 657 - 682

... noncooperatively [39–43]. Much progress has been achieved in the mathematical theory of MFG for time-dependent problems [13,23–25,27,31,34,47,48] and for stationary problem...

Computer Systems Organization and Communication Networks | Mean-field games | Monotone schemes | Numerical methods | Economic Theory/Quantitative Economics/Mathematical Methods | Communications Engineering, Networks | Mathematics | Operations Research, Management Science | Game Theory, Economics, Social and Behav. Sciences | Economics, general | EXISTENCE | CONGESTION | WEAK KAM THEORY | EQUATIONS | PDE METHODS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | APPROXIMATION SCHEME | DENSITY CONSTRAINTS | SYSTEMS | 1ST-ORDER

Computer Systems Organization and Communication Networks | Mean-field games | Monotone schemes | Numerical methods | Economic Theory/Quantitative Economics/Mathematical Methods | Communications Engineering, Networks | Mathematics | Operations Research, Management Science | Game Theory, Economics, Social and Behav. Sciences | Economics, general | EXISTENCE | CONGESTION | WEAK KAM THEORY | EQUATIONS | PDE METHODS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | APPROXIMATION SCHEME | DENSITY CONSTRAINTS | SYSTEMS | 1ST-ORDER

Journal Article

Journal of the American Mathematical Society, ISSN 0894-0347, 09/2008, Volume 21, Issue 3, pp. 615 - 669

.... These subsets are generalizations of Lagrangian graphs, we call them pseudographs. They emerge in a natural way from Fathi's weak KAM theory...

Integers | Equivalence relation | Mathematical manifolds | Mathematical theorems | Mathematical functions | Trajectories | Mathematics | Mathematical minima | Lagrangian function | Continuous functions | Arnold's diffusion | Weak KAM | Hamilton-Jacobi equation | Mather sets | MATHEMATICS | MECHANISM | weak KAM | CONNECTING ORBITS | LAGRANGIAN SYSTEMS | DIFFUSION | TIME | Dynamical Systems

Integers | Equivalence relation | Mathematical manifolds | Mathematical theorems | Mathematical functions | Trajectories | Mathematics | Mathematical minima | Lagrangian function | Continuous functions | Arnold's diffusion | Weak KAM | Hamilton-Jacobi equation | Mather sets | MATHEMATICS | MECHANISM | weak KAM | CONNECTING ORBITS | LAGRANGIAN SYSTEMS | DIFFUSION | TIME | Dynamical Systems

Journal Article

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