Weak KAM Theory with a Non-Smooth Lagrangian

Weak KAM theory emerged as an attempt to describe the solutions of the calculus of variations problem over long time intervals. It aims to find a function $\phi : M \to \mathbb{R}$ and a constant $\bar{H}$ such that the solution to the calculus of variations problem
to minimize
$$ \phi(x(T)) + \int_0^T L(x(t), \dot{x}(t))dt $$

subject to $x(0) = y$ on an arbitrary interval $[0,T]$ is $\phi(y) - \bar{H}T$. Here, the value $\bar{H}$ is called the effective Hamiltonian or Mañé's critical value. Within the framework of weak KAM theory, it is shown that the function $\phi$ and the constant $\bar{H}$ satisfy a Hamilton-Jacobi type equation in the viscosity sense
$$ H(x, -\nabla\phi) = -\bar{H}, $$
where $H$ is the Hamiltonian:
$$ H(x, p) = \max_{v \in T_x M} [p \cdot v - L(x,v)]. $$
Note that in this case, the solution to the calculus of variations problem
to minimize
$$ \frac{1}{T} \int_0^T L(x(t), \dot{x}(t))dt $$

converges to $\bar{H}$ as $T \to \infty$.
Weak KAM theory was initially developed under the assumption of smoothness and coercivity of the Lagrangian $L$. At the same time, the weak KAM HJ equation has a solution on the torus in the case where $L$ does not satisfy the smoothness condition.
In this regard, it is of interest to develop elements of weak KAM theory in the case of a non-smooth Lagrangian, based on the viscosity solution of the weak KAM HJ equation. This report is devoted to this issue. In doing so, we make extensive use of the method of shifting along the proximal subgradient, proposed by F. Clarke, Yu.S. Ledyaev, and A.I. Subbotin.