Optimality Conditions for Smooth Monge Solutions of the Monge–Kantorovich problem

The Monge–Kantorovich problem (MKP) with given marginals defined on closed domains $X\subset\mathbb{R}^n$, $Y\subset\mathbb{R}^m$ and a smooth cost function $c\colon X\times Y\to\mathbb{R}$ is considered. Conditions are obtained (both necessary ones and sufficient ones) for the optimality of a Monge solution generated by a smooth measure-preserving map $f\colon X\to Y$. The proofs are based on an optimality criterion for a general MKP in terms of nonemptiness of the sets $Q_0(\zeta)=\{u\in\mathbb{R}^X:u(x)-u(z)\le\zeta(x,z)$ for all $x,z\in X\}$ for special functions $\zeta$ on $X\times X$ generated by $c$ and $f$. Also, earlier results by the author are used when considering the above-mentioned nonemptiness conditions for the case of smooth $\zeta$.