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Equivalence of ray monotonicity properties and classification of optimal transport maps for strictly convex norms

  • Ping Chen EMAIL logo
Published/Copyright: September 3, 2020
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 15 Issue 3

Abstract

In this paper, we first define ray increasing and decreasing monotonicity of maps. If 𝑇 is an optimal transport map for the Monge problem with cost function ∥ y - x ∥ sc in R n or 𝑇 is an optimal transport map for the Monge problem with cost function d ⁢ ( x , y ) , the geodesic distance, in more general, non-branching geodesic spaces 𝑋, we show respectively equivalence of some previously introduced monotonicity properties and the property of ray increasing as well as ray decreasing monotonicity which we define in this paper. Then, by solving secondary variational problems associated with strictly convex and concave functions respectively, we show that there exist ray increasing and decreasing optimal transport maps for the Monge problem with cost function ∥ y - x ∥ sc . Finally, we give the classification of optimal transport maps for the Monge problem such that the cost function ∥ y - x ∥ sc further satisfies the uniform smoothness and convexity estimates. That is, all of the optimal transport maps for such Monge problem can be divided into three different classes: the ray increasing map, the ray decreasing map and others.

Keywords: Classification; equivalence; optimal transport map; Monge problem
MSC 2010: 49J45; 49Q20; 49K30

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11601193

Funding statement: The research of the author was supported by the National Natural Science Foundation of China (No. 11601193), the Qing Lan Project of Jiangsu Province, and Jiangsu Overseas Visiting Scholar program for University Prominent Young & Middle-aged Teachers and Presidents.

  1. Communicated by: Frank Duzaar

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Received: 2019-11-12
Revised: 2020-07-30
Accepted: 2020-08-19
Published Online: 2020-09-03
Published in Print: 2022-07-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Michell truss type theories as a Γ-limit of optimal design in linear elasticity
  3. The local structure of the free boundary in the fractional obstacle problem
  4. Homogenization of quadratic convolution energies in periodically perforated domains
  5. Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow
  6. Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth
  7. On a comparison principle for Trudinger’s equation
  8. Equivalence of ray monotonicity properties and classification of optimal transport maps for strictly convex norms
  9. An extensive study of the regularity of solutions to doubly singular equations
  10. New features of the first eigenvalue on negatively curved spaces
  11. High order curvature flows of plane curves with generalised Neumann boundary conditions
  12. (BV,L p )-decomposition, p = 1,2, of functions in metric random walk spaces
  13. Causal variational principles in the σ-locally compact setting: Existence of minimizers
  14. On sub-Riemannian geodesic curvature in dimension three
  15. Existence for singular critical exponential (𝑝, 𝑄) equations in the Heisenberg group
https://doi.org/10.1515/acv-2019-0099
Keywords for this article
Classification; equivalence; optimal transport map; Monge problem

Articles in the same Issue

  1. Frontmatter
  2. Michell truss type theories as a Γ-limit of optimal design in linear elasticity
  3. The local structure of the free boundary in the fractional obstacle problem
  4. Homogenization of quadratic convolution energies in periodically perforated domains
  5. Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow
  6. Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth
  7. On a comparison principle for Trudinger’s equation
  8. Equivalence of ray monotonicity properties and classification of optimal transport maps for strictly convex norms
  9. An extensive study of the regularity of solutions to doubly singular equations
  10. New features of the first eigenvalue on negatively curved spaces
  11. High order curvature flows of plane curves with generalised Neumann boundary conditions
  12. (BV,L p )-decomposition, p = 1,2, of functions in metric random walk spaces
  13. Causal variational principles in the σ-locally compact setting: Existence of minimizers
  14. On sub-Riemannian geodesic curvature in dimension three
  15. Existence for singular critical exponential (𝑝, 𝑄) equations in the Heisenberg group
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