Abstract
We prove a local regularity (and a corresponding a priori estimate) for plurisubharmonic solutions of the nondegenerate complex Monge–Ampère equation assuming that their W 2, p-norm is under control for some p > n(n − 1). This condition is optimal. We use in particular some methods developed by Trudinger and an estimate for the complex Monge–Ampère equation due to Kołodziej.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bedford E.: Survey of pluri-potential theory. In: Fornæss, J.E. (ed.) Several Complex Variables. Proceedings of the Mittag-Leffler Institute, vol. 1987–1988, Princeton University Press, New Jersy (1993)
Bedford E., Taylor B.A.: The Dirichlet problem for a complex Monge–Ampère equation. Invent. Math. 37, 1–44 (1976)
Błocki Z.: The complex Monge–Ampère operator in hyperconvex domains. Ann. Scuola Norm. Sup. Pisa 23, 721–747 (1996)
Błocki, Z.: On the regularity of the complex Monge–Ampère operator. In: Complex Geometric Analysis in Pohang, 1997. Contemp. Math., vol. 222, pp. 181–189. Amer. Math. Soc., Providence (1999)
Błocki Z.: Interior regularity of the complex Monge–Ampère equation in convex domains. Duke Math. J. 105, 167–181 (2000)
Cegrell U., Persson L.: The Dirichlet problem for the complex Monge–Ampère operator: stability in L 2. Michigan Math. J. 39, 145–151 (1992)
He, W.: On regularity of complex Monge–Ampère equations. arXiv:1002.4825v2
Kołodziej S.: Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge–Ampère operator. Ann. Pol. Math. 65, 11–21 (1996)
Kołodziej S.: The complex Monge–Ampère equation. Acta Math. 180, 69–117 (1998)
Pogorelov, A.V.: The Dirichlet problem for the multidimensional analogue of the Monge–Ampère equation. Dokl. Akad. Nauk SSSR 201, 790–793 (1971) [transl.: Soviet Math. Dokl. 12, 1727–1731 (1971)]
Siu Y.-T.: Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics. Birkhäuser, Basel (1987)
Trudinger N.S.: Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations. Invent. Math. 61, 67–79 (1980)
Trudinger N.S.: Regularity of solutions of fully nonlinear elliptic equations. Boll. Un. Mat. Ital. A (6) 3, 421–430 (1984)
Urbas J.: Regularity of generalized solutions of Monge–Ampère equations. Math. Z. 197, 365–393 (1988)
Acknowledgments
Part of the research was done while the second named author was visiting the Princeton University. He would like to thank this institution for the perfect working conditions and hospitality and especially professor Gang Tian for his encouragement and help.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the projects N N201 2683 35 and 189/6 PR EU/2007/7 of the Polish Ministry of Science and Higher Education.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Błocki, Z., Dinew, S. A local regularity of the complex Monge–Ampère equation. Math. Ann. 351, 411–416 (2011). https://doi.org/10.1007/s00208-010-0609-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-010-0609-0