Asymptotic Lipschitz regularity of viscosity solutions of Hamilton-Jacobi equations

Abstract:

For each continuous initial data $\varphi (x)\in C(M,\mathbb {R})$, we obtain the asymptotic Lipschitz regularity of the viscosity solution of the following evolutionary Hamilton-Jacobi equation with convex and coercive Hamiltonians: \begin{equation*} \begin {cases} \partial _tu(x,t)+H(x,\partial _xu(x,t))=0,\\ u(x,0)=\varphi (x). \end{cases} \end{equation*}