Skip to main content
Log in

On the Regularity of Optimal Transportation Potentials on Round Spheres

Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satisfied on the round sphere, when the cost-function is the geodesic distance squared. In order to generalise Loeper’s calculation to a broader class of cost-functions, the (A3) condition is reformulated via a stereographic projection that maps charts of the sphere into Euclidean space. This reformulation subsequently allows one to verify the (A3) condition for any case where the cost-function of the associated optimal transportation problem can be expressed as a function of the geodesic distance between points on a round sphere. With this, several examples of such cost-functions are then analysed to see whether or not they satisfy this (A3) condition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. Caffarelli, L.A.: Allocation maps with general cost functions. In: Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math., vol. 177, pp. 29–35. Dekker, New York (1996)

    Google Scholar 

  2. Delanoë, P., Loeper, G.: Gradient estimates for potentials of invertible gradient-mappings on the sphere. Calc. Var. Partial Differ. Equ. 26(3), 297–311 (2006)

    Article  MATH  Google Scholar 

  3. Evans, L.C.: Partial differential equations and Monge-Kantorovich mass transfer (2001). http://math.berkeley.edu/~evans/Monge-Kantorovich.survey.pdf

  4. Fathi, A., Figalli, A.: Optimal transportation on non-compact manifolds. Isr. J. Math. 175, 1–59 (2010). doi:10.1007/s11856-010-0001-5

    Article  MathSciNet  MATH  Google Scholar 

  5. Figalli, A., Rifford, L.: Continuity of optimal transport maps and convexity of injectivity domains on small deformations of \(\Bbb{S}^{2}\). Commun. Pure Appl. Math. 62(12), 1670–1706 (2009). doi:10.1002/cpa.20293.

    Article  MathSciNet  MATH  Google Scholar 

  6. Figalli, A., Rifford, L., Villani, C.: Nearly Round Spheres Look Convex. Am. J. Math. 134(1), 109–139 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gangbo, W., McCann, R.J.: Optimal maps in Monge’s mass transport problem. C. R. Acad. Sci. Paris, Sér. I Math. 321(12), 1653–1658 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Kantorovich, L.: On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS (N. S.) 37, 199–201 (1942)

    MathSciNet  Google Scholar 

  9. Kantorovich, L.V.: On a problem of Monge. Usp. Mat. Nauk 3, 225–226 (1948) (in Russian)

    Google Scholar 

  10. Kim, Y.H., McCann, R.J.: Continuity, curvature, and the general covariance of optimal transportation. J. Eur. Math. Soc. 12(4), 1009–1040 (2010). doi:10.4171/JEMS/221

    Article  MathSciNet  MATH  Google Scholar 

  11. Loeper, G.: On the regularity of solutions of optimal transportation problems. Acta Math. 202(2), 241–283 (2009). doi:10.1007/s11511-009-0037-8

    Article  MathSciNet  MATH  Google Scholar 

  12. Loeper, G.: Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna. Arch. Rational Mech. Anal. 199(1), 269–289 (2010). doi:10.1007/s00205-010-0330-x

    Article  MathSciNet  Google Scholar 

  13. Ma, X.N., Trudinger, N.S., Wang, X.J.: Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177(2), 151–183 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. McCann, R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(3), 589–608 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Monge, G.: Mémoire sur la théorie des déblais et des remblais. In: Historie de l’Académie Royale des Sciences de Paris, pp. 666–704 (1781)

    Google Scholar 

  16. von Nessi, G.T.: Regularity results for potential functions of the optimal transportation problem on spheres and related Hessian equations. Ph.D. thesis, Australian National University (2008)

  17. Trudinger, N.S., Wang, X.J.: On the second boundary value problem for Monge-Ampère type equations and optimal transportation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(1), 143–174 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Urbas, J.: Mass Transfer Problems. Lecture notes given at the University of Bonn (1998)

    Google Scholar 

  19. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Greg T. von Nessi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

von Nessi, G.T. On the Regularity of Optimal Transportation Potentials on Round Spheres. Acta Appl Math 123, 239–259 (2013). https://doi.org/10.1007/s10440-012-9764-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-012-9764-5

Keywords

Mathematics Subject Classification (2010)

Navigation