Dirichlet problems for Monge-Ampère equation degenerate on boundary (Q1192391)

The author proves the existence of a unique convex globally smooth solution of the Dirichlet problem \(\text{det} D^ 2u=K(x)\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) in a smooth strictly convex domain \(\Omega\) in \(R^ 2\) under the assumption that \(K\) is a smooth positive function with \(K=0\) and \(DK\neq0\) on \(\partial\Omega\). This result is interesting primarily in view of the fact that the vanishing of \(K\) on \(\partial\Omega\) forces the equation to become degenerate there. Up till now there were no results on the existence of globally smooth solutions in this case. Many of the arguments are similar to those used in the nondegenerate case. The main new idea is the derivation of a modulus of continuity estimate for the second derivatives on \(\partial\Omega\). Higher order estimates then follow from a general regularity theorem for degenerate Monge-Ampère equations due to the author and \textit{C. Zuily} [Commun. Partial Diff. Equations 16, No. 6/7, 997-1031 (1991; Zbl 0742.35009)].