Free boundaries in optimal transport and Monge-Ampere obstacle problems

By Luis Caffarelli, Robert J. McCann. 2010

239 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b1e0d66147a53a666f23bf43021dca2fa892e8b9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Hölder Continuity and Injectivity of Optimal Maps (2011, background): “This question is not merely of technical interest, since discontinuities in f± arise unavoidably in applications such as partial transport problems [10] [3] [15] [16].”
Scaling Algorithms for Unbalanced Transport Problems (2016, background): “If F3 is the indicator of equality with a positive real number, this algorithm solves the partial optimal transport problem [19, 33] but more general F3 can be considered such as, for instance, a range constraint on the total mass.”
Multi-marginal optimal transport: theory and applications (2014, background): “In the former variant, uniqueness of solutions for infimal convolution type costs has been established with Kitagawa [48], under a natural and necessary condition on the marginals, extending work in the two marginal case of Caffarelli–McCann [11] and Figalli [30].”
A General Duality Theorem for the Monge--Kantorovich Transport Problem (2009, background): “ the value of the 1−εpartial transportation problem Pε := inf ˆ hc,πi = Z X×Y c(x,y)dπ(x,y) : π∈ Πε(µ,ν) ˙ (6) . This partial transport problem has recently been studied by L. Caﬀarelli and R. McCann [CM06] as well as A. Figalli [Fig09]. In their work the emphasis is on a ﬁner analysis of the 1or, m”
Unbalanced optimal transport: Dynamic and Kantorovich formulations (2015, background): “It is proved in [6] that for any choice of δ > 0 corresponds the choice of m ( δ ) such that 0 ≤ m ( δ ) ≤ min { ρ 0 (Ω) , ρ 1 (Ω) } for which the minimizers of (5.3) and the minimizers of are the same.”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, methodology): “This method is quite flexible: it has the advantage of yielding certain conclusions under weaker assumptions on the data, and has therefore proven useful for addressing such phenomena as the free boundary which arises in partial transport problems, where the densities f± need not be continuous and a”
The Optimal Partial Transport Problem (2010, background): “1 regularity of the free boundaries away from Ω∩Λ are shown.”
Iterative Bregman Projections for Regularized Transportation Problems (2014, background): “They show in particular that if there exists an hyperplane separating the support of the two marginals then the “active region” is separated from the “inactive region” by a free boundary which can be parameterized as a semi concave graph over the separating hyperplane.”
Global regularity for the Monge-Ampère equation with natural boundary condition (2018, methodology): “These new techniques may apply to other problems related to Monge-Ampère type equations, such as the regularity of the free boundary problem in optimal transportation in dimension two [8], which is currently under investigations by the authors.”