Interior regularity of the complex Monge-Ampère equation in convex domains
A well known theorem proved by \textit{E. Bedford} and \textit{B. A. Taylor} [Invent. Math. 37, 1-44 (1976; Zbl 0315.31007)] asserts that: If \(\Omega\) is a strictly pseudoconvex bounded domain in \(\mathbb{C}^n\), \(\psi\) is a nonnegative, continuous function on \(\overline\Omega\) and \(f\) a continuous function on \(\partial\Omega\), then there exists a unique function \(u\) psh on \(\Omega\), continuous on \(\overline\Omega\), satisfying \[ Mu= \psi\quad\text{on }\Omega,\qquad u= f\quad\text{on }\partial\Omega, \] where \(M\) is the Monge-Ampère operator (\(M(u)= \text{det}({\partial^2u\over\partial z_i\partial\overline z_j})\) if \(u\) is \(C^2\)). Many regularity results for \(u\) are known in this setting. This existence theorem with some regularity was generalized by the author in [Mich. Math. J. 42, 269-275 (1995; Zbl 0841.35017)] when \(\Omega\) is a hyperconvex domain provided that \(f\) can be extended continuously to a psh function on \(\Omega\) \((*)\). In this paper the author proves that \(u\) is \(C^\infty\), when \(\Omega\) is a bounded convex domain, \(f= 0\), \(\psi> 0\), \(\psi\) is \(C^\infty\) and \(|D\psi^{1/n}|\) is bounded. In fact it is sufficient to show that, for \(\Omega'\subset\subset\Omega\), there exists \(\alpha\in (0,1)\) such that \(u\in C^{2,\alpha}(\Omega')\). This is proved using an estimate for \(|Du|\) and an already known estimate for \(|\Delta u|\) proved by \textit{F. Schulz} [J. Reine Angew. Math. 348, 88-93 (1984; Zbl 0521.35010)]. The first one is the only point needing the convexity of \(\Omega\), and the author conjectures that the result remains valid for \(\Omega\) hyperconvex. In a last section the author investigates the special case when \(\Omega\) is a polydisc \(P\) and \(f\) is \(C^{1,1}\) on \(\partial P\), subharmonic on every analytic disc embedded in \(\partial P\) (which implies \((*)\)). He proves then that \(u\) is \(C^\infty\) when \(\psi> 0\), \(\psi\) is \(C^\infty\) and \(|D^2\psi^{1/n}|\) is bounded. He also gives an example showing that this regularity fails for \(\psi= 0\).
scientific article; zbMATH DE number 1820758
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