Caffarelli, Luis A. The regularity of mappings with a convex potential. (English) Zbl 0753.35031 J. Am. Math. Soc. 5, No. 1, 99-104 (1992). Let \(\Omega_ 1\) and \(\Omega_ 2\) be two bounded domains in \(\mathbb{R}^ n\) and \(f_ 1,f_ 2\) two nonnegative real functions defined, respectively, in \(\Omega_ 1,\Omega_ 2\). Consider the Monge-Ampère equation \[ f_ 2(\nabla\psi)\text{det}D_{ij}\psi=f_ 1(X).\tag{1} \] Under some additional assumptions, Y. Brenier [C. R. Acad. Sci., Paris, Sér. I 305, 805-808 (1987; Zbl 0652.26017)] has proved existence and uniqueness for (1).In the present work the author proves that if \(\Omega_ 2\) is convex and \(f_ i\), \(1/f_ i\) (\(i=1,2\)) are bounded, then Brenier’s solution is a weak solution in the sense of Alexandrov (i.e., \(\text{det}D_{ij}\psi\) has no singular part and \(\psi\) is strictly convex). He also shows that \(\psi\) is \(C^{1,\beta}\) for some \(\beta\). The proof uses some techniques developed by the author [Ann. Math., II. Ser. 131, No. 1, 129- 134 (1990; Zbl 0704.35045)]. Reviewer: A.Cañada (Granada) Cited in 6 ReviewsCited in 147 Documents MathOverflow Questions: Bounding probability densities on a Wasserstein-2 geodesic MSC: 35J60 Nonlinear elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations 35B65 Smoothness and regularity of solutions to PDEs Keywords:Monge-Ampère equation; weak solution Citations:Zbl 0652.26017; Zbl 0704.35045 PDFBibTeX XMLCite \textit{L. A. Caffarelli}, J. Am. Math. Soc. 5, No. 1, 99--104 (1992; Zbl 0753.35031) Full Text: DOI References: [1] Yann Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 19, 805 – 808 (French, with English summary). · Zbl 0652.26017 [2] L. A. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math. (2) 131 (1990), no. 1, 129 – 134. · Zbl 0704.35045 [3] Luis A. Caffarelli, Interior \?^{2,\?} estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135 – 150. · Zbl 0704.35044 [4] Luis A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 965 – 969. · Zbl 0761.35028 [5] A. V. Pogorelov, Monge-Ampère equations of elliptic type, Translated from the first Russian edition by Leo F. Boron with the assistance of Albert L. Rabenstein and Richard C. Bollinger, P. Noordhoff, Ltd., Groningen, 1964. · Zbl 0133.04902 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.