Variational principle for contact Hamiltonian systems and its applications

Kaizhi Wang, Lin Wang, Jun Yan*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

45 Citations (Scopus)

Abstract

In [8], the authors provided an implicit variational principle for the contact Hamilton's equations {x˙=[Formula presented](x,u,p),p˙=−[Formula presented](x,u,p)−[Formula presented](x,u,p)p,(x,p,u)∈TM×R,u˙=[Formula presented](x,u,p)⋅p−H(x,u,p), where M is a closed, connected and smooth manifold and H=H(x,u,p) is strictly convex, superlinear in p and Lipschitz in u. In the present paper, we focus on two applications of the variational principle: 1. We provide a representation formula for the solution semigroup of the evolutionary equation wt(x,t)+H(x,w(x,t),wx(x,t))=0; 2. We study the ergodic problem of the stationary equation via the solution semigroup. More precisely, we find pairs (u,c) with u∈C(M,R) and c∈R which, in the viscosity sense, satisfy the stationary partial differential equation H(x,u(x),ux(x))=c.

Original languageEnglish
Pages (from-to)167-200
Number of pages34
JournalJournal des Mathematiques Pures et Appliquees
Volume123
DOIs
Publication statusPublished - Mar 2019
Externally publishedYes

Keywords

  • Contact Hamilton's equations
  • First-order PDEs
  • Implicit variational principle
  • Viscosity solutions

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