On the second boundary value problem for Monge-Ampère type equations and optimal transportation

By Neil S. Trudinger, Xu-jia Wang. 2006

202 citations recorded by the source corpus.

This paper is concerned with the existence of globally smooth so- lutions for the second boundary value problem for certain Monge-Amp` ere type equations and the application to regularity of potentials in optimal transportation. In particular we address the fundamental issue of determining conditions on costs and domains to ensure that optimal mappings are smooth diffeomorphisms. The cost functions satisfy a weak form of the condition (A3), which was introduced in a recent paper with Xi-nan Ma, in conjunction with interior regularity. Our condition is optimal and includes the quadratic cost function case of Caffarelli and Urbas as well as the various examples in our previous work. The approach is through the derivation of global estimates for second derivatives of solutions.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f431e931c4d91790bec73eea1f09276ea226b94c

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Citations in the corpus

Pseudo-Riemannian geometry calibrates optimal transportation (2009, background): “We define the mass of space-like currents in spaces with indefinite metrics.”
Hölder regularity of optimal mappings in optimal transportation (2009, background): “In the papers [18, 19 ,26], the mass distributions are assumed to be C 1,1 smooth and positive.”
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].”
Free boundaries in optimal transport and Monge-Ampere obstacle problems (2010, background): “Regularity results for non-quadratic costs are a very recent development even in the context of the fixed boundary (complete transfer) problem, where Ma, Trudinger & Wang have identified a concavity condition on the Hessian of the cost, which — for smooth data and suitable domain geometry — yields r”
Oblique boundary value problems for augmented Hessian equations I (2015, methodology): “They play the same role as the function |x|2 in the standard Hessian case and replace the barriers arising from the “A-boundedness” conditions introduced for Monge-Ampère type equations in [16, 18].”
Regularity of optimal transport in curved geometry: The nonfocal case (2010, unknown): “) Conversely, smoothness results have been obtained under various sets of assumptions including either the weak or the strong MTW condition [10, 16, 19, 20, 22, 27]; such results are reviewed in [30, Chapter 12].”
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].”
Hölder Continuity and Injectivity of Optimal Maps (2011, methodology): “If the source density f(x) is bounded away from zero and infinity in an open region U ′ ⊂ R, and the target density f(y) is bounded away from zero and infinity on its support V ⊂ R, which is strongly c-convex with respect to U , and the transportation cost c satisfies the (A3)w condition of Trudinge”
On the Dirichlet problem for Monge-Ampère type equations (2013, methodology): “These were subsequently extended to global estimates and regularity in [28] under a weak form of this condition called A3w; (see [23] for further discussion).”
Continuity and injectivity of optimal maps for non-negatively cross-curved costs (2009, methodology): “The present paperalso provides an argument required by Figalli and Loeper to on lude in two dimensions ontinuity of optimal maps under the weaker (in fa t, ne essary) hypothesis (A3w)[17℄.”
Interior C 2,α Regularity for Potential Functions in Optimal Transportation (2009, methodology): “Our proof uses Pogorelov’s approach for the standard Monge–Ampère equation, as for the global estimate in [18], but the details are much more complicated.”
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].”
C1 regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two (2009, background): “Under Aw, classical C 2 and higher regularity holds for smooth positive data [7] [10], while assuming the stronger condition As, C 1,α regularity holds for rough and possibly vanishing Monge-Ampère measures [5].”
Regularity of Optimal Maps on the Sphere: the Quadratic Cost and the Reflector Antenna (2013, methodology): “This paragraph is a short reminder of the results established in [16,17, 22 ].”
On the regularity of solutions of optimal transportation problems (2009, methodology): “paper [ 21 ], they could lower this condition down to Aw (condition A3w in their paper) to obtain a sufficient condition for globalC 2 (and subsequently C ∞ ) regularity, assuming uniform”
Optimal transportation, topology and uniqueness (2010, background): “Adding further restrictions beyond this twist hypothesis allowed Ma, Trudinger, Wang [72] [100], and later Loeper [68], to develop a regularity theory for the map f1 : X −→ Y , embracing Delanoë [36], Caffarelli [20] [21] and Urbas’ [102] results for the quadratic cost, Gangbo and McCann’s for its ”