scholarly journals On smooth time functions

2011 ◽  
Vol 152 (2) ◽  
pp. 303-339 ◽  
Author(s):  
ALBERT FATHI ◽  
ANTONIO SICONOLFI
Keyword(s):  
Kam Theory ◽  
Time Function ◽  
Lorentzian Manifolds ◽  
Weak Kam Theory ◽  
Continuity Properties ◽  
Integral Curves ◽  
Intrinsic Length ◽  
Definition Of ◽  
General Abstract ◽  
Jacobi Equations

AbstractWe are concerned with the existence of smooth time functions on connected time-oriented Lorentzian manifolds. The problem is tackled in a more general abstract setting, namely in a manifold M where is just defined a field of tangent convex cones (Cx)x ∈ M enjoying mild continuity properties. Under some conditions on its integral curves, we will construct a time function.Our approach is based on the definition of an intrinsic length for curves indicating how a curve is far from being an integral trajectory of Cx. We find connections with topics pertaining to Hamilton–Jacobi equations, and make use of tools and results issued from weak KAM theory.

Systems of Weakly Coupled Hamilton-Jacobi Equations with Implicit Obstacles

2012 ◽  
Vol 64 (6) ◽  
pp. 1289-1309 ◽  
Author(s):  
Diogo Gomes ◽  
António Serra
Keyword(s):  
Dirichlet Problem ◽  
Viscosity Solutions ◽  
Existence And Uniqueness ◽  
Kam Theory ◽  
Optimal Switching ◽  
Uniqueness Of Solutions ◽  
Weak Kam Theory ◽  
Weakly Coupled ◽  
Aubry Set ◽  
Jacobi Equations

Abstract In this paper we study systems of weakly coupled Hamilton-Jacobi equations with implicit obstacles that arise in optimal switching problems. Inspired by methods from the theory of viscosity solutions and weak KAM theory, we extend the notion of Aubry set for these systems. This enables us to prove a new result on existence and uniqueness of solutions for the Dirichlet problem, answering a question of F. Camilli, P. Loreti, and N. Yamada.

scholarly journals Weak KAM Approach to First-Order Mean Field Games with State Constraints

2021 ◽  
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng ◽  
Cristian Mendico ◽  
Kaizhi Wang
Keyword(s):  
Time Horizon ◽  
State Constraints ◽  
Kam Theory ◽  
Mean Field ◽  
Mean Field Games ◽  
Asymptotic Behavior Of Solutions ◽  
Weak Kam Theory ◽  
First Order ◽  
Jacobi Equations ◽  
Ergodic Mean

AbstractWe study the asymptotic behavior of solutions to the constrained MFG system as the time horizon T goes to infinity. For this purpose, we analyze first Hamilton–Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on [0, T] converges to the solution of the ergodic system as T goes to infinity.

Some Results on Weak KAM Theory for Time-periodic Tonelli Lagrangian Systems

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Kaizhi Wang ◽  
Yong Li
Keyword(s):  
Kam Theory ◽  
Lagrangian Systems ◽  
Specific Class ◽  
Initial Condition ◽  
Commun Math Phys ◽  
Arbitrary Continuous Function ◽  
Weak Kam Theory ◽  
Definition Of ◽  
Time Periodic ◽  
Math Phys

AbstractThis paper contributes several results on weak KAM theory for time-periodic Tonelli Lagrangian systems. Wang and Yan [Commun. Math. Phys. 309 (2012), 663-691] introduced a new kind of Lax-Oleinik type operator associated with any time-periodic Tonelli Lagrangian. Firstly, using the new operator we give an equivalent definition of the backward weak KAM solution. Then we prove a result on the asymptotic behavior of the new operators with an arbitrary continuous function as initial condition, by taking advantage of the definition mentioned above. Finally, for a specific class of time-periodic Tonelli Lagrangians, we discuss the rate of convergence of the new operators.

Weak KAM Theory and Forcing Equivalence

2020 ◽  
pp. 55-65
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Kam Theory ◽  
Alternative Definition ◽  
Weak Kam Theory ◽  
Peierls Barrier ◽  
Definition Of ◽  
Standard Presentation ◽  
Alpha Function

This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to allow different periods in the t component. The chapter points out an alternative definition of the alpha function, namely, one can replace the class of minimal measures with the class of closed measures. It then considers a dual setting which corresponds to forward dynamic. It also looks at elementary solutions, static classes, and Peierls barrier. In many parts of the proof, the chapter studies the hyperbolic property of a minimizing orbit, for which the concept of Green bundles is very useful.

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