On the twist condition and $c$-monotone transport plans
Thierry Champion Luigi De Pascale
Discrete & Continuous Dynamical Systems - A 2014, 34(4): 1339-1353 doi: 10.3934/dcds.2014.34.1339
A usual approach for proving the existence of an optimal transport map, be it in ${\mathbb R}^d$ or on more general manifolds, involves a regularity condition on the transport cost (the so-called Left Twist condition, i.e. the invertibility of the gradient in the first variable) as well as the fact that any optimal transport plan is supported on a cyclically-monotone set. Under the classical assumption that the initial measure does not give mass to sets with $\sigma$-finite $\mathcal{H}^{d-1}$ measure and a stronger regularity condition on the cost (the Strong Left Twist), we provide a short and self-contained proof of the fact that any feasible transport plan (optimal or not) satisfying a $c$-monotonicity assumption is induced by a transport map. We also show that the usual costs induced by Tonelli Lagrangians satisfy the Strong Left Twist condition we propose.
keywords: Tonelli Lagrangian. cyclical monotonicity optimal transport problem Monge-Kantorovich problem
Giuseppe Buttazzo Luigi De Pascale Ilaria Fragalà
Discrete & Continuous Dynamical Systems - A 2007, 18(1): 219-220 doi: 10.3934/dcds.2007.18.219
keywords: distances Finsler metrics.
Topological equivalence of some variational problems involving distances
Giuseppe Buttazzo Luigi De Pascale Ilaria Fragalà
Discrete & Continuous Dynamical Systems - A 2001, 7(2): 247-258 doi: 10.3934/dcds.2001.7.247
To every distance $d$ on a given open set $\Omega\subseteq\mathbb R^n$, we may associate several kinds of variational problems. We show that, on the class of all geodesic distances $d$ on $\Omega$ which are bounded from above and from below by fixed multiples of the Euclidean one, the uniform convergence on compact sets turns out to be equivalent to the $\Gamma$-convergence of each of the corresponding variational problems under consideration.
keywords: geodesic distances Gamma-convergence length functionals.

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