Regularization of subsolutions in discrete weak KAM theory (Q2841816)

This article continues a series of the second author works on the existence of subsolutions in discrete weak KAM theory [\textit{M. Zavidovique}, J. Mod. Dyn. 4, No. 4, 693--714 (2010; Zbl 1211.37077), Comment. Math. Helv. 87, No. 1, 1--39 (2012; Zbl 1275.37031)]. Given a (cost) function \(c: M \times M \rightarrow \mathbb{R}\), \(M\) a smooth and complete Riemannian manifold, a subsolution is a function \(u: M \rightarrow \mathbb{R}\) that satisfies \(\forall (x,y) \in M \times M, \;\;u(y)-u(x) \leq c(x,y)\).NEWLINENEWLINEThe authors provide a proof of \(C^{1,1}\) subsolutions by using Lax-Oleinik operators. This method resembles the proof given by \textit{P. Bernard} for the case of Tonelli Lagrangians on compact manifolds [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 3, 445--452 (2007; Zbl 1133.35027)]. In that case, the Lax-Oleinik semi-groups preserve subsolutions, the image of a continuous function by the backward (foward) semi-group is semi-concave (resp. semi-convex). Hence the composition of the forward and the backward pair, for small positive values \(t\) and \(s\), of a continuous function is both semi-concave and semi-convex, therefore it is \(C^{1,1}\).NEWLINENEWLINEIn the discrete setting the Lax-Oleinik operators associated to a cost function \(c\) are defined as follows: \(T_{c}^{-} u(x)= \inf_{y \in M} (u(y)+c(y,x))\) and \(T_{c}^{+} u(x)= \sup_{y \in M} (u(y)-c(x,y))\).NEWLINENEWLINEThe main result of the article is the following.NEWLINENEWLINETheorem: Suppose that for each subsolution \(u\), the functions \(T_{c}^{-} u\) and \(T_{c}^{+} u\) are locally semi-concave (Hypothesis 1), then the set of locally \(C^{1,1}\) subsolutions is dense in the set of continuous solutions for the Whitney topology in \(C^{0}(M,\mathbb{R})\).NEWLINENEWLINEThe authors describe two settings where the above Hypothesis 1 holds, one of them was used by the second author in a previous work in order to prove an analogous result. As in the case of Tonelli Lagrangians, the authors define an Aubry subset \(\mathcal{A}\) and prove, under the above Hypothesis 1, the existence of a locally \(C^{1,1}\) subsolution that is free and smooth in the complement of \(\mathcal{A}\).