Hölder Continuity and Injectivity of Optimal Maps

By Alessio Figalli, Young-Heon Kim, Robert J. McCann. 2011

76 citations recorded by the source corpus.

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x, y). If the source density f+(x) is bounded away from zero and infinity in an open region $${U' \subset \mathbf{R}^n}$$, and the target density f−(y) is bounded away from zero and infinity on its support $${\overline{V} \subset \mathbf{R}^n}$$, which is strongly c-convex with respect to U′, and the transportation cost c satisfies the $${\mathbf{A3}_{\rm w}}$$ condition of Trudinger and Wang (Ann Sc Norm Super Pisa Cl Sci 5, 8(1):143–174, 2009), we deduce the local Hölder continuity and injectivity of the optimal map inside U′ (so that the associated potential u belongs to $${C^{1,\alpha}_{loc}(U')}$$). Here the exponent α > 0 depends only on the dimension and the bounds on the densities, but not on c. Our result provides a crucial step in the low/interior regularity setting: in a sequel (Figalli et al., J Eur Math Soc (JEMS), 1131–1166, 2013), we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. Three key tools are introduced in the present paper. Namely, we first find a transformation that under $${\mathbf{A3}_{\rm w}}$$ makes c-convex functions level-set convex (as was also obtained independently from us by Liu (Calc Var Partial Diff Eq 34:435–451, 2009)). We then derive new Alexandrov type estimates for the level-set convex c-convex functions, and a topological lemma showing that optimal maps do not mix the interior with the boundary. This topological lemma, which does not require $${\mathbf{A3}_{\rm w}}$$, is needed by Figalli and Loeper (Calc Var Partial Diff Eq 35:537–550, 2009) to conclude the continuity of optimal maps in two dimensions. In higher dimensions, if the densities f± are Hölder continuous, our result permits continuous differentiability of the map inside U′ (in fact, $${C^{2,\alpha}_{loc}}$$ regularity of the associated potential) to be deduced from the work of Liu et al. (Comm Partial Diff Eq 35(1):165–184, 2010).

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/a7729b880897452edbd18ef9270fdcc1719a91a2

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Citations in the corpus

On the continuity of center-outward distribution and quantile functions (2018, background): “The argument in this case is inspired by [10].”
The Monge–Ampère equation and its link to optimal transportation (2013, result): “ses MTW(0). Hence, it would be nice to have a Holder regularity theorem which includes at the same time Caarelli’s and Loeper’s results. As proved by the second author together with Kim and McCann [53], such a result is indeed true (see also [55, 66, 117]): Theorem 4.8. Let c: X Y !R satisfy (C0)-(C”
Partial regularity for optimal transport maps (2012, background): “More precisely, if the MTW condition holds (together with some suitable convexity assumptions on the target domain), then the optimal map is smooth [35, 36, 21, 30, 19].”
NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS ON RIEMANNIAN MANIFOLDS (2011, result): “Actually, as can easily be seen from the proof of the above result, by the results in [11] the following general statement (already present in the proof of [12, Corollary 5.”