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Boundary regularity of maps with convex potentials. II. (English) Zbl 0916.35016

Suppose \(\Omega_1\) and \(\Omega_2\) are two bounded domains in \(\mathbb{R}^n\) with \(| \partial\Omega_1| =| \partial\Omega_2| =0\), and \(f,g\) are functions on \(\Omega_1\) and \(\Omega_2\) respectively, bounded away from zero and infinity and satisfying \(\int_{\Omega_1}f = \int_{\Omega_2}g\). Then, as shown by Y. Brenier [Commun. Pure Appl. Math. 44, No. 4, 375-417 (1991; Zbl 0738.46011)], there are convex potentials \(\psi\) and \(\phi\) such that \(\nabla\psi:\Omega_1\rightarrow\Omega_2\) and \(\nabla\phi:\Omega_2\rightarrow\Omega_1\) are surjective maps in the a.e. sense, and \(\psi\) satisfies the Monge-Ampère equation \[ g(\nabla\psi)\det\nabla^2\psi = f(x) \] in the integral sense that \[ \int_{\Omega_2}\eta(y)g(y)dy = \int_{\Omega_1}\eta(\nabla\psi)f(x)dx \] for any continuous \(\eta\) on \(\mathbb{R}^n\). In two previous papers [J. Am. Math. Soc. 5, No. 1, 99-104 (1992; Zbl 0753.35031); Commun. Pure Appl. Math. 45, No. 9, 1141-1151 (1992; Zbl 0778.35015)] the author proved the interior and global \(C^{1,\alpha}\) regularity of \(\psi,\phi\) under convexity assumptions on one or both of the domains. Here he carries this further by proving global \(C^{2,\alpha}\) regularity if \(f,g\) are \(C^{0,\alpha}\) up to the boundary and \(\Omega_1,\Omega_2\) are strictly convex with \(C^2\) boundaries. A similar result was proved by Ph. Delanoë [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8, No. 5, 443-457, (1991; Zbl 0778.35037)] in the two-dimensional case, and very recently also by the reviewer [J. Reine Angew. Math. 487, 115-124 (1997; Zbl 0880.35031)] in all dimensions using different techniques.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35J60 Nonlinear elliptic equations
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