Geodesic
Sobolev regularity for the Monge–Ampère equation, with application to the semigeostrophic equations
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 103-118 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this note we review some recent results on the Sobolev regularity of solutions to the Monge–Ampère equation, and show how these estimates can be used to prove some global existence results for the semigeostrophic equations. 
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Alessio Figalli. Sobolev regularity for the Monge–Ampère equation, with application to the semigeostrophic equations. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 103-118. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a5/

[1] L. Ambrosio, M. Colombo, G. De Philippis, A. Figalli, “Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case”, Comm. Partial Differential Equations (to appear)

[2] L. Ambrosio, M. Colombo, G. De Philippis, A. Figalli, “A global existence result for the semigeostrophic equations in three dimensional convex domains”, Discrete Contin. Dyn. Syst. (to appear)

[3] J.-D. Benamou, Y. Brenier, “Weak existence for the semigeostrophic equation formulated as a coupled Monge–Ampère/transport problem”, SIAM J. Appl. Math., 58:5 (1998), 1450–1461 | DOI | MR | Zbl

[4] L. Caffarelli, “A localization property of viscosity solutions to the Monge–Ampère equation and their strict convexity”, Ann. of Math. (2), 131:1 (1990), 129–134 | DOI | MR | Zbl

[5] L. Caffarelli, “Interior $W^{2,p}$ estimates for solutions of the Monge–Ampère equation”, Ann. of Math. (2), 131:1 (1990), 135–150 | DOI | MR | Zbl

[6] L. Caffarelli, “Some regularity properties of solutions to Monge–Ampère equations”, Comm. Pure Appl. Math., 44 (1991), 965–969 | DOI | MR | Zbl

[7] L. Caffarelli, C. Gutierrez, “Real analysis related to the Monge–Ampère equation”, Trans. Amer. Math. Soc., 348:3 (1996), 1075–1092 | DOI | MR | Zbl

[8] D. Cordero Erausquin, “Sur le transport de mesures périodiques”, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 199–202 | DOI | MR | Zbl

[9] M. Cullen, A mathematical theory of large-scale atmosphere/ocean flow, Imperial College Press, 2006

[10] M. Cullen, M. Feldman, “Lagrangian solutions of semigeostrophic equations in physical space”, SIAM J. Math. Anal., 37 (2006), 1371–1395 | DOI | MR | Zbl

[11] G. De Philippis, A. Figalli, “$W^{2,1}$ regularity for solutions of the Monge–Ampère equation”, Invent. Math., 192:1 (2013), 55–69 | DOI | MR | Zbl

[12] G. De Philippis, A. Figalli, O. Savin, “A note on interior $W^{2,1+\epsilon}$ estimates for the Monge–Ampère equation”, Math. Ann. (to appear)

[13] C. Gutierrez, The Monge–Ampére equation, Progress in Nonlinear Differential Equations and their Applications, 44, Birkhäuser Boston, Inc., Boston, MA, 2001 | MR | Zbl

[14] C. Gutierrez, Q. Huang, “Geometric properties of the sections of solutions to the Monge–Ampére equation”, Trans. Amer. Math. Soc., 352:9 (2000), 4381–4396 | DOI | MR | Zbl

[15] G. Loeper, “On the regularity of the polar factorization for time dependent maps”, Calc. Var. Partial Differential Equations, 22 (2005), 343–374 | DOI | MR | Zbl

[16] T. Schmidt, $W^{2,1+\epsilon}$ estimates for the Monge–Ampère equation, Preprint, 2012 | MR

[17] X.-J. Wang, “Some counterexamples to the regularity of Monge–Ampère equations”, Proc. Amer. Math. Soc., 123:3 (1995), 841–845 | MR | Zbl