C1 regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two

By Alessio Figalli, Grégoire Loeper. 2009

71 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/96c2aa0955b55bf8e728b71c80cc9cc4fcceabab

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Continuity and injectivity of optimal maps for non-negatively cross-curved costs (2009, methodology): “The on lusion then follows from Loeper'slo al to global prin iple, Corollary 3.4 above.”
Hölder Continuity and Injectivity of Optimal Maps (2011, methodology): “rem 5.1 does not require (A3) w, however. Let us point out that, in two dimensions, there is an alternate approach to establishing continuity of optimal maps; it was carried out by Figalli and Loeper [20] following Alexandrov’s strategy [2], but their result relies on our Theorem 5.1. 2 Main result ”
Continuity, curvature, and the general covariance of optimal transportation (2007, background): “These include Hölder continuity results for maps between rough measures which optimize weakly regular costs in two dimensions [18] and continuity results for such maps for non-negatively crosscurved costs in higher dimensions [17], regularity of optimal maps on strictly regular Riemannian manifolds ”
On the Ma–Trudinger–Wang curvature on surfaces (2010, background): “The Ma–Trudinger–Wang (MTW) tensor is a nonlocal generalization of sectional curvature, involving fourth-order derivatives of the squared distance function [4, 7, 9, 10, 11, 18, 22, 25, 26, 27, 28, 29].”
Interior C 2,α Regularity for Potential Functions in Optimal Transportation (2009, unknown): “By a related approximation argument, in [13] Loeper also proved (A3w) is necessary for regularity, namely if c does not satisfy (A3w), there exist smooth, positive mass distributions such that the potential function is not C1 smooth.”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “ optimal transportation and the regularity theory for general Monge-Ampere type equations. (1)As we already mentioned, in dimension 2 stronger regularity results for solutions of (1.1) are available [3, 22, 55]. In particular, Alexandrov showed in [3] that uis continuous dierentiable assuming only t”
NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS ON RIEMANNIAN MANIFOLDS (2011, background): “…positive bounded densities on a connected Riemannian manifold (M, g), and let ψ : M → R be a locally semiconvex c-convex function such that the map T : M →M given by T (x) = expx ( ∇ψ(x) ) a.e. x ∈M. satisfies T#µ = ν (observe that in this setting we cannot talk about optimal maps, since µ may…”
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 [19, 21, 30, 35, 36].”
Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2 (2009, background): “The above definition extends the definition of the Ma-Trudinger-Wang tensor, which was first introduced in [21] and extensively studied in [11, 12, 13, 17, 18, 19, 20, 22, 23, 29].”