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

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 Added by Lin Wang
 Publication date 2014
  fields
and research's language is English




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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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