Hölder continuous solutions to Monge-Ampère equations. (English) Zbl 1157.32033

Guedj, V.; Kolodziej, S.; Zeriahi, A.

Bull. Lond. Math. Soc. 40, No. 6, 1070-1080 (2008).

Summary: We study the regularity of solutions to the Dirichlet problem for the complex Monge-Ampère equation \((dd^c u)^n=f dV\) on a bounded strongly pseudoconvex domain \({\Omega} \subset \mathbb C^n\). We show, under a mild technical assumption, that the unique solution \(u\) to this problem is Hölder continuous if the boundary data \({\phi}\) is Hölder continuous and the density \(f\) belongs to \(L^p({\Omega})\) for some \(p>1\). This improves previous results by Bedford and Taylor and Kolodziej.

MSC:

32W20	Complex Monge-Ampère operators
32U15	General pluripotential theory