Assume that M is a closed, connected and smooth Riemannian manifold. The authors consider the evolutionary Hamilton-Jacobi equation \(\begin{cases}\partial_{t}u(x,t)+H(x,u(x,t),\partial_{x}u(x,t))=0, & (x,t) \in M \times (0, +\infty), \\u(x,0)=\varphi(x), &\end{cases}\) where φ ∈ C(M) and the stationary one \(H(x,u(x),\partial_{x}u(x))=0,\) where H(x,u,p) is continuous, convex and coercive in p, uniformly Lipschitz in u. By introducing a solution semigroup, the authors provide a representation formula of the viscosity solution of the evolutionary equation. As its applications, they obtain a necessary and sufficient condition for the existence of the viscosity solutions of the stationary equations. Moreover, they prove a new comparison theorem depending on the neighborhood of the projected Aubry set essentially, which is different from the one for the Hamilton-Jacobi equation independent of u.
