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.


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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