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

Albert Fathi

doi:10.1017/S0308210550000064

Abstract

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.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Kong, Yuedong and Xu, Junxiang 2015. Weak KAM theorem for Hamilton‐Jacobi equations with Neumann boundary conditions on noncompact manifolds. Mathematical Methods in the Applied Sciences, Vol. 38, Issue. 14, p. 2974.

Davini, A. and Zavidovique, M. 2015. On the (non) existence of viscosity solutions of multi-time Hamilton–Jacobi equations. Journal of Differential Equations, Vol. 258, Issue. 2, p. 362.

Barles, Guy and Meireles, Joao 2016. On unbounded solutions of ergodic problems in ℝmfor viscous Hamilton–Jacobi equations. Communications in Partial Differential Equations, Vol. 41, Issue. 12, p. 1985.

Davini, Andrea Fathi, Albert Iturriaga, Renato and Zavidovique, Maxime 2016. Convergence of the solutions of the discounted Hamilton–Jacobi equation. Inventiones mathematicae, Vol. 206, Issue. 1, p. 29.

Davini, Andrea and Siconolfi, Antonio 2016. Existence and regularity of strict critical subsolutions in the stationary ergodic setting. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Vol. 33, Issue. 2, p. 243.

Dabkowski, Michael G. and Lock, Michael T. 2018. On the proportionality of Chern and Riemannian scalar curvatures. Geometriae Dedicata, Vol. 195, Issue. 1, p. 57.

GOMES, Diogo A. MITAKE, Hiroyoshi and TRAN, Hung V. 2018. The selection problem for discounted Hamilton–Jacobi equations: some non-convex cases. Journal of the Mathematical Society of Japan, Vol. 70, Issue. 1,

Chen, Cui Cheng, Wei and Zhang, Qi 2018. Lasry–Lions approximations for discounted Hamilton–Jacobi equations. Journal of Differential Equations, Vol. 265, Issue. 2, p. 719.

Chen, Qinbo Cheng, Wei Ishii, Hitoshi and Zhao, Kai 2019. Vanishing contact structure problem and convergence of the viscosity solutions. Communications in Partial Differential Equations, Vol. 44, Issue. 9, p. 801.

Roos, Valentine 2019. Variational and viscosity operators for the evolutionary Hamilton–Jacobi equation. Communications in Contemporary Mathematics, Vol. 21, Issue. 04, p. 1850018.

Cannarsa, Piermarco Cheng, Wei and Fathi, Albert 2021. Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry. Publications mathématiques de l'IHÉS, Vol. 133, Issue. 1, p. 327.

Fathi, Albert 2022. Urysohn-type theorem under a dynamical constraint: Non-compact case. Journal of Modern Dynamics, Vol. 18, Issue. 0, p. 523.

Zavidovique, Maxime 2022. Convergence of solutions for some degenerate discounted Hamilton–Jacobi equations. Analysis & PDE, Vol. 15, Issue. 5, p. 1287.

Shu, Xiang Yan, Jun and Zhao, Kai 2022. Viscosity subsolutions of Hamilton–Jacobi equations and invariant sets of contact Hamilton systems. Journal of Mathematical Physics, Vol. 63, Issue. 10,

Chen, Qinbo 2023. Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function. Advances in Calculus of Variations, Vol. 16, Issue. 1, p. 45.

Fathi, Albert and Pageault, Pierre 2023. Aubry Set on Infinite Cyclic Coverings. Regular and Chaotic Dynamics, Vol. 28, Issue. 4-5, p. 425.

Capuzzo Dolcetta, Italo and Davini, Andrea 2023. On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case. Communications in Partial Differential Equations, Vol. 48, Issue. 4, p. 576.

Chen, Qinbo Fathi, Albert Zavidovique, Maxime and Zhang, Jianlu 2024. Convergence of the solutions of the nonlinear discounted Hamilton–Jacobi equation: The central role of Mather measures. Journal de Mathématiques Pures et Appliquées, Vol. 181, Issue. , p. 22.

Article contents

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

Abstract

Access options

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

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

Abstract

Access options

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests