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Regularity of generalized solutions of Monge-Ampère equations. (English) Zbl 0617.35017

We study the interior regularity of convex generalized solutions of Monge-Ampère equations of the form det \(D^ 2 u=f(x,u,Du)\) in \(\Omega\), where \(\Omega\) is a bounded convex domain in \({\mathbb{R}}^ n\) and \(f\in C^{1,1}(\Omega \times {\mathbb{R}}\times {\mathbb{R}}^ n)\) is a positive function. In general, sufficient initial regularity of u in \(\Omega\), or of \(\partial \Omega\) and \(u| _{\partial \Omega}\), is required to conclude that u is a classical solution. Our main theorem asserts that if u is continuous on \({\bar \Omega}\), and \(\partial \Omega\) and \(u| _{\partial \Omega}\) are of class \(C^{1,\alpha}\) for some \(\alpha >1- 2/n\), then \(u\in C^{3,\beta}(\Omega)\) for all \(\beta <1\). This generalizes earlier work of A. V. Pogorelov and others, and an example of Pogorelov shows that if \(\partial \Omega\) and \(u| _{\partial \Omega}\) are only of class \(C^{1,1-2/n}\), then u need not be a classical solution, even for positive analytic f.
We also prove estimates of strict convexity and related regularity theorems for solutions u belonging to suitable Hölder or Sobolev spaces, and also to more general function spaces which allow us to give sharp results.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35G20 Nonlinear higher-order PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
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