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\(C^ 2\) a priori estimates for degenerate Monge-Ampère equations. (English) Zbl 0879.35059
The author studies the degenerate case of the Dirichlet problem for the Monge-Ampère equation \[ \det(D^2u)= f\geq 0\quad\text{in }\Omega\subset\mathbb R^n,\quad u|_{\partial\Omega}=\phi. \] \(C^{1,1}\)-regularity and estimates are known in the case \(f\equiv 0\) [see N. S. Trudinger and J. Urbas, Bull. Aust. Math. Soc. 30, 321–334 (1984; Zbl 0557.35054)] or if \(f= g^n\) with some \(C^{1,1}\)-function \(g\) [see L. Caffarelli, J. J. Kohn, L. Nirenberg and J. Spruck, Commun. Pure Appl. Math. 38, 209–252 (1985; Zbl 0598.35048)].
The author gives a new sufficient condition for \(f\), which allows him to prove \(C^{1,1}\)-estimates in the case \(\phi=0\). This condition is fulfilled e.g. for \(f=|g|\), \(g\in C^{1,1}\) if \(n=2\), or \(f\in C^{3,1}\) if \(n=3\). Also a nonvanishing boundary value is allowed, if. e.g. \(f\) is nondegenerate near the boundary. The technique is further extended to the equation of prescribed Gaussian curvature.
Reviewer: M.Wiegner (Aachen)

MSC:
35J70 Degenerate elliptic equations
35B45 A priori estimates in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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