2 problems

Let $M$ be a compact manifold, let $H\in C^2(T^*M)$ be a Tonelli Hamiltonian, and let $u$ be a weak KAM solution of … where $c$ is Mañé's critical value. Write $\mathrm{Cut}(u)$ fo…

Common critical subsolution conjecture. If $[H_1,H_2]=0$, then there exists at least one common $C^{1,1}$ subsolution for