Regularization of Subsolutions in Discrete Weak KAM Theory

Patrick Bernard; Maxime Zavidovique

doi:10.4153/cjm-2012-059-3

Regularization of Subsolutions in Discrete Weak KAM Theory

Canadian Journal of Mathematics ◽

10.4153/cjm-2012-059-3 ◽

2013 ◽

Vol 65 (4) ◽

pp. 740-756 ◽

Cited By ~ 3

Author(s):

Patrick Bernard ◽

Maxime Zavidovique

Keyword(s):

Kam Theory ◽

Weak Kam Theory ◽

Aubry Set

AbstractWe expose different methods of regularizations of subsolutions in the context of discrete weak KAM theory that allow us to prove the existence and the density of C1,1 subsolutions. Moreover, these subsolutions can be made strict and smooth outside of the Aubry set.

Download Full-text

Systems of Weakly Coupled Hamilton-Jacobi Equations with Implicit Obstacles

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-085-7 ◽

2012 ◽

Vol 64 (6) ◽

pp. 1289-1309 ◽

Cited By ~ 4

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.

Download Full-text

Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210550000064 ◽

2012 ◽

Vol 142 (6) ◽

pp. 1193-1236 ◽

Cited By ~ 9

Author(s):

Albert Fathi

Keyword(s):

Prior Knowledge ◽

Jacobi Equation ◽

Kam Theory ◽

General Setting ◽

Point Of View ◽

Weak Kam Theory ◽

First Order ◽

Aubry Set ◽

The Subject ◽

Hamilton Jacobi Equation

We introduce the notion of a viscosity solution for the first-order Hamilton–Jacobi equation, in the more general setting of manifolds, to obtain a weak KAM theory using only tools from partial differential equations. This work should be accessible to people with no prior knowledge of the subject.

Download Full-text