Essential Forward Weak KAM Solution for the Convex Hamilton—Jacobi Equation

for a convex, coercive continuous Hamiltonian on closed Riemannian manifold M, we construct unique forward weak KAM solution of $$H(x,{d_x}u) = c(H)$$ by vanishing discount approach, where c(H) is the Mañé critical value. We also discuss dynamical significance such special solution.