Weak KAM theory for general Hamilton-Jacobi equations II: the fundamental solution under Lipschitz conditions

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)=ϕ(x), \end{cases} \end{equation*} where $ϕ(x)\in C(M,\mathbb{R})$. Under some assumptions on the convexity of $H(x,u,p)$ with respect to $p$ and the uniform Lipschitz of $H(x,u,p)$ with respect to $u$, we establish a variational principle and provide an intrinsic relation between viscosity solutions and certain minimal characteristics. By introducing an implicitly defined {\it fundamental solution}, we obtain a variational representation formula of the viscosity solution of the evolutionary Hamilton-Jacobi equation. Moreover, we discuss the large time behavior of the viscosity solution of the evolutionary Hamilton-Jacobi equation and provide a dynamical representation formula of the viscosity solution of the stationary Hamilton-Jacobi equation with strictly increasing $H(x,u,p)$ with respect to $u$.

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