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Bulletin of the London Mathematical Society Advance Access originally published online on November 4, 2008
Bulletin of the London Mathematical Society 2008 40(6):1070-1080; doi:10.1112/blms/bdn092
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© 2008 London Mathematical Society

Hölder continuous solutions to Monge–Ampère equations

V. Guedj and S. Kolodziej

Laboratoire Emile Picard
UMR 5580
Université Paul Sabatier
118 route de Narbonne
31062 Toulouse cedex 04
France
zeriahi@picard.ups-tlse.fr

A. Zeriahi

Institute of Mathematics
Jagiellonian University
Reymonta 4
30-059 Krakow
Poland
Slawomir.Kolodziej@im.uj.edu.pl

Received 26 October 2007. Revision received 22 July 2008.

We study the regularity of solutions to the Dirichlet problem for the complex Monge–Ampère equation (ddc u)n=f dV on a bounded strongly pseudoconvex domain {Omega}subCn. 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 Lp({Omega}) for some p>1. This improves previous results by Bedford and Taylor and Kolodziej.

2000 Mathematics Subject Classification 32W20, 32U15.


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