Weak KAM theory for  HAMILTON-JACOBI equations depending on unknown functions

Xifeng  Su; Lin  Wang; Jun  Yan

doi:10.3934/dcds.2016080

Article Contents

2016, Volume 36, Issue 11: 6487-6522. Doi: 10.3934/dcds.2016080

This issue Previous Article Backward iteration algorithms for Julia sets of Möbius semigroups Next Article On a constant rank theorem for nonlinear elliptic PDEs

Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions

1.
School of Mathematical Sciences, Beijing Normal University, Beijing 100875
2.
Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084
3.
School of Mathematical Sciences, Fudan University, Shanghai Key Laboratory for Contemporary Applied Mathematics, Shanghai 200433

Received: August 31, 2015

Revised: June 30, 2016

Published: May 2017

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Abstract

We consider the evolutionary Hamilton-Jacobi equation depending on the unknown function with the continuous initial condition on a connected closed manifold. Under certain assumptions on $H(x,u,p)$ with respect to $u$ and $p$, we provide an implicit variational principle. By introducing an implicitly defined solution semigroup and an admissible value set $\mathcal{C}_H$, we extend weak KAM theory to certain more general cases, in which $H$ depends on the unknown function $u$ explicitly. As an application, we show that for $0\notin \mathcal{C}_H$, as $t\rightarrow +\infty$, the viscosity solution of \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\varphi(x), \end{cases} \end{equation*} diverges, otherwise for $0\in \mathcal{C}_H$, it converges to a weak KAM solution of the stationary Hamilton-Jacobi equation \begin{equation*} H(x,u(x),\partial_xu(x))=0. \end{equation*}

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Mathematics Subject Classification: Primary: 37J50, 35F21; Secondary: 35D40.

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