Abstract
In this paper, we establish the global $C^{2,\alpha }$ and $W^{2,p}$ regularity for the Monge-Ampère equation ${\mathrm{det}}\, D^2u = f$ subject to boundary condition $Du(\Omega ) = \Omega ^*$, where $\Omega $ and $\Omega ^*$ are bounded convex domains in the Euclidean space $\mathbb{R}^n$ with $C^{1,1}$ boundaries, and $f$ is a Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.
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pages = {115--124},
issn = {0075-4102},
mrclass = {35J65},
mrnumber = {1454261},
mrreviewer = {Jia Xing Hong},
doi = {10.1515/crll.1997.487.115},
url = {https://doi.org/10.1515/crll.1997.487.115},
zblnumber = {0880.35031},
} -
@BOOK{V1,
author = {Villani, Cédric},
title = {Topics in Optimal Transportation},
series = {Grad. Stud. Math.},
volume = {58},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2003},
pages = {xvi+370},
isbn = {0-8218-3312-X},
mrclass = {90-02 (28D05 35B65 35J60 49N90 49Q20 90B20)},
mrnumber = {1964483},
doi = {10.1090/gsm/058},
url = {https://doi.org/10.1090/gsm/058},
zblnumber = {1106.90001},
} -
@BOOK{V2,
author = {Villani, Cédric},
title = {Optimal Transport, Old and New},
series = {Grundlehren Math. Wissen.},
note = {},
publisher = {Springer-Verlag, Berlin},
year = {2009},
pages = {xxii+973},
isbn = {978-3-540-71049-3},
mrclass = {49-02 (28A75 37J50 49Q20 53C23 58E30)},
mrnumber = {2459454},
mrreviewer = {Dario Cordero-Erausquin},
doi = {10.1007/978-3-540-71050-9},
url = {https://doi.org/10.1007/978-3-540-71050-9},
zblnumber = {1156.53003},
}