Let $\mathbb{T}^d$ be the $d$-dimensional torus. Let $L:\mathbb{T}^d\times\mathbb{R}^d \mapsto \mathbb{R}$ be a Lagrangian $L(x,v)$, smooth in both variables, strictly convex in the velocity $v$, and coercive, that is, $$\lim_{|v|\rightarrow\infty}\inf_{x}\frac{L(x,v)}{|v|}=\infty .$$ The Hamiltonian $H$ is the Legendre transform of $L$ given by $$H(p,x) = \sup_{v}(p\cdot v-L(x,v)),$$ which is strictly convex and coercive as a function of $p$.
Let $u:\mathbb{T}^d\mapsto\mathbb{R}$ be a viscosity solution of the Hamilton-Jacobi equation $H(D_xu(x),x)=\bar{H}$, where $\bar{H}$ is a constant. Note that $u$ is a classical solution in points of differentiablity.
I am trying to prove that $$|D_xH(D_xu^{\epsilon}(x+h)-D_xH(D_xu(x))|\leq C|D_xu^{\epsilon}(x+h)-D_xu(x)|,$$ that is a Lipschitz condition, but I am stuck.
I proved that $u$ is Lipschitz, bounded and differentiable almost evereywhere (with respect to Lebesgue measure). I know that H is smooth, so $D_xH$ is continuous. For example, if I could prove that $D_xH$ it is differentiable with respect to $p$, then its derivative is bounded and so, $D_xH$ would be Lipschitz, but I am not sure one can do this. So I think I have to use continuity and some properties of $H$, $u$ and $\mathbb{T}^d$.
Note that $$ \frac{|D_xH(p,x) - D_xH(q,x)|}{|x-y|}|x-y| \leq |x-y|\sup |D_p (D_x H)|. $$ If we have that $D_p (D_x H)$ es bounded we are done.