Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Hamilton-Jacobi Equations and Weak KAM Theory

  • Antonio SiconolfiEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_268-3

Definition of the Subject

This entry aims to illustrate some applications of weak KAM theory to the analysis of Hamilton-Jacobi equations. The presentation focuses on two specific problems, namely, the existence of C1 classical subsolutions for a class of stationary (i.e., independent of the time) Hamilton-Jacobi equations and the long-time behavior of viscosity solutions of an evolutive version of it.

The Hamiltonian is assumed to satisfy mild regularity conditions, under which the corresponding Hamilton equations cannot be written. Consequently, PDE techniques will be solely employed in the analysis, since the powerful tools of the Hamiltonian dynamics are not available.


Given a continuous or more regular Hamiltonian H( x, p) defined on the cotangent bundle of a boundaryless manifold M, where x and p are the state and the momentum variable, respectively, and satisfying suitable convexity and coercivity assumptions, is considered the family of Hamilton-Jacobi equations


Viscosity Solution Critical Curve Critical Equation Viscosity Subsolution Critical Curf 
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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Dip. di Matematica“La Sapienza”; Università di RomaRomeItaly