Geodesic
Sobolev regularity for the Monge--Amp\`ere 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

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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. 
@article{ZNSL_2013_411_a5,
     author = {Alessio Figalli},
     title = {Sobolev regularity for the {Monge--Amp\`ere} equation, with application to the semigeostrophic equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {103--118},
     publisher = {mathdoc},
     volume = {411},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a5/}
}
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Alessio Figalli. Sobolev regularity for the Monge--Amp\`ere 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/