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REGULARITY OF MONGE–AMPÈRE EQUATIONS IN OPTIMAL TRANSPORTATION

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  27 January 2011

JIAKUN LIU
JIAKUN LIU*
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA (email: jiakunl@math.princeton.edu)
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Abstract

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Keywords

Monge–Ampère equationoptimal transportation

MSC classification

Secondary: 35J60: Nonlinear elliptic equations
Type
Abstracts of Australasian PhD Theses
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 1 , February 2011 , pp. 173 - 176
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Caffarelli, L. A., ‘Interior W 2,p estimates for solutions of the Monge–Ampère equation’, Ann. of Math. (2) 131 (1990), 135–150.CrossRefGoogle Scholar
[2]Caffarelli, L. A., ‘Some regularity properties of solutions of Monge Ampère equation’, Comm. Pure Appl. Math. 44 (1991), 965–969.CrossRefGoogle Scholar
[3]Caffarelli, L. A., ‘The regularity of mappings with a convex potential’, J. Amer. Math. Soc. 5 (1992), 99–104.Google Scholar
[4]Caffarelli, L. A., ‘Non linear elliptic theory and the Monge–Ampère equation’, Proceedings of the ICM, Vol. 1 (Beijing, 2002) (Higher Ed. Press, Beijing, 2002), pp. 179–187.Google Scholar
[5]Delanoë, Ph., ‘Classical solvability in dimension two of the second boundary value problem associated with the Monge–Ampère operator’, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 443–457.CrossRefGoogle Scholar
[6]Jian, H. Y. and Wang, X.-J., ‘Continuity estimates for the Monge–Ampère equation’, SIAM J. Math. Anal. 39 (2007), 608–626.CrossRefGoogle Scholar
[7]Liu, J., ‘Hölder regularity of optimal mappings in optimal transportation’, Calc. Var. Partial Differential Equations 34 (2009), 435–451.CrossRefGoogle Scholar
[8]Liu, J., Trudinger, N. S. and Wang, X.-J., ‘On asymptotic behaviour and W 2,p regularity of potentials in optimal transportation’, submitted.Google Scholar
[9]Liu, J., Trudinger, N. S. and Wang, X.-J., ‘Interior C 2,α regularity for potential functions in optimal transportation’, Comm. Partial Differential Equations 35 (2010), 165–184.CrossRefGoogle Scholar
[10]Loeper, G., ‘On the regularity of maps solutions of optimal transportation problems’, Acta Math. 202 (2009), 241–283.Google Scholar
[11]Ma, X. N., Trudinger, N. S. and Wang, X.-J., ‘Regularity of potential functions of the optimal transportation problem’, Arch. Ration. Mech. Anal. 177 (2005), 151–183.CrossRefGoogle Scholar
[12]Urbas, J., ‘On the second boundary value problem of Monge–Ampère type’, J. reine angew. Math. 487 (1997), 115–124.Google Scholar
[13]Villani, C., Topics in Optimal Transportation, Graduate Studies in Mathematics, 58 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
[14]Wang, X.-J., ‘Remarks on the regularity of Monge–Ampère’, in: Proc. Inter. Conf. Nonlinear PDE, (eds. Dong, G. C. and Lin, F. H.) (Academic Press, Beijing, 1992), pp. 257–263.Google Scholar
[15]Wang, X.-J., ‘Schauder estimate for elliptic and parabolic equations’, Chin. Ann. Math. Ser. B 27 (2006), 637–642.CrossRefGoogle Scholar