Essential forward weak KAM solution for the convex Hamilton-Jacobi equation

The paper proves that the forward discounted solutions u^+_λ exist uniquely, are equi-bounded/Lipschitz, converge (as λ→0^+) to a unique forward 0–weak KAM solution u^+_0, and provides the dual characterizations u^+_0(x)=inf{w(x): w≺L+c(H), ∫w dμ≥0 ∀μ∈M} and u^+_0(x)=−inf_{μ∈M}∫h_∞(x,y)dμ(y) (Theorem 11), relying on a symmetry trick Ĥ(x,p)=H(x,−p), the equality of projected Mather measures M̂=M, and the identity ĥ_∞(y,x)=h_∞(x,y) (Proposition/Lemma 21), plus the DFIZ discounted-limit results for Ĥ; see Theorem 11, Proposition 22, Lemma 31–32, and Lemma 21 in the paper . The candidate solution establishes the same results but via a different route: it builds the discounted forward Lax–Oleinik operator, shows it is a strict contraction, obtains a unique fixed point u^+_λ and its domination/calibration properties, derives equi-regularity and Arzelà–Ascoli convergence to u^+_0, and then proves the same two characterizations. Minor gaps in the candidate’s exposition (e.g., independence of the fixed point from the choice of t and integrating the PDE against Mather measures for non-C^1 u^+_λ) can be patched with standard arguments or by appealing to the DFIZ framework cited by the paper. Hence both are correct, with substantially different proof strategies.