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arXiv · 1408.3790

Weak KAM theory for general Hamilton-Jacobi equations III: the variational principle under Osgood conditions

Lin Wang·Jun Yan
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Abstract

We consider the following evolutionary Hamilton-Jacobi equation with initial condition: \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x), \end{cases} \end{equation*} where $\phi(x)\in C(M,\mathbb{R})$. Under some assumptions on the convexity of $H(x,u,p)$ with respect to $p$ and the Osgood growth of $H(x,u,p)$ with respect to $u$, we establish an implicitly variational principle and provide an intrinsic relation between viscosity solutions and certain minimal characteristics. Moreover, we obtain a representation formula of the viscosity solution of the evolutionary Hamilton-Jacobi equation.

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Lin Wang, Jun Yan. 2014-08-17. Weak KAM theory for general Hamilton-Jacobi equations III: the variational principle under Osgood conditions. https://arxiv.org/abs/1408.3790

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