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On the second boundary value problem for equations of Monge-Ampère type. (English) Zbl 0880.35031

The author proves, by using the continuity method, the existence of globally smooth convex solutions of Monge-Ampère equations of the form \(\text{det }D^2u= f(x,u,Du)\) in \(\Omega\) subject to the boundary condition \(Du(\Omega)=\Omega^*\) where \(\Omega\) and \(\Omega^*\) are smooth uniformly convex domain in \(\mathbb{R}^n\). Under a weaker hypotheses on \(f,\Omega, \Omega^*\), L. Caffarelli [Ann. Math., II. Ser. 144, 453-496 (1996)] proved a similar result. However, the techniques are different.
Reviewer: Ma Li (Beijing)

MSC:

35G30 Boundary value problems for nonlinear higher-order PDEs
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