On Strict Convexity and Continuous Differentiability of Potential Functions in Optimal Transportation

By Neil S. Trudinger, Xu-jia Wang. 2009

89 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/d38618ccf43f5127c47251268d9d93abf6dd3f74

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In Applied Mathematics

Citations in the corpus

On the Dirichlet problem for Monge-Ampère type equations (2013, methodology): “From [27], we know that an elliptic solution will be c-convex in Ω if Ω is c-convex with respect to each y ∈ R, that is the images cy(·, y)(Ω) are convex in R .”
Continuity, curvature, and the general covariance of optimal transportation (2007, methodology): “This new global result illustrates an advantage of our method over other approaches [42], [43], [31], [32], which would require a regularity result for optimal maps (or some a priori estimates) to obtain the conclusion of Theorem 3.”
Interior C 2,α Regularity for Potential Functions in Optimal Transportation (2009, methodology): “We remark that in [19], only the boundedness of 1/ ∗ from above is needed.”
Regularity of optimal transport in curved geometry: The nonfocal case (2010, unknown): “Various forms of the Ma–Trudinger–Wang condition 9 3.”
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].”
STABILITY OF A 4TH-ORDER CURVATURE CONDITION ARISING IN OPTIMAL TRANSPORT THEORY (2008, background): “There is by now plenty of evidence that these conditions play a key role in the regularity theory of optimal transport [10, 12, 13, 14, 15, 17, 19, 20].”
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, 23, 24, 28].”
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].”