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The regularity of mappings with a convex potential. (English) Zbl 0753.35031

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)].

MSC:

35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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[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 · doi:10.2307/1971509
[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 · doi:10.2307/1971510
[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 · doi:10.1002/cpa.3160440809
[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
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