## 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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### References:

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