×

The complex Monge-Ampère operator in hyperconvex domains. (English) Zbl 0878.31003

Let \(\Omega\) be a hyperconvex domain in \(\mathbb{C}^n\), i.e. any boundary point of \(\Omega\) admits a weak plurisubharmonic barrier. The author considers such domains and proves the following result.
Let \(\Omega\) be as above, suppose that \(f\in C(\partial\Omega)\) can be continuously extended to a plurisubharmonic function on \(\Omega\), and let \(F\) be continuous up to \(\Omega\), \(F\geq 0\). Then there exists a plurisubharmonic function \(u\in C(\overline{\Omega})\) such that \[ \text{det}\Biggl( \frac{\partial^2\psi} {\partial z_j\partial\overline{z}_k} \Biggr)=F \] and \(u|_{\partial\Omega}=f\).

MSC:

31C10 Pluriharmonic and plurisubharmonic functions
32W20 Complex Monge-Ampère operators
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] E. Bedford , Survey of pluri-potential theory, Several Complex Variables, Proceedings of the Mittag-Leffler Institute, 1987-1988 , J.E. Fornæss (ed.), Princeton Univ. Press , 1993 . MR 1207855 | Zbl 0786.31001 · Zbl 0786.31001
[2] E. Bedford - B.A. Taylor , The Dirichlet problem for a complex Monge-Ampère equation , Invent. Math. 37 ( 1976 ), 1 - 44 . MR 445006 | Zbl 0315.31007 · Zbl 0315.31007 · doi:10.1007/BF01418826
[3] E. Bedford - B.A. Taylor , A new capacity for plurisubharmonic functions , Acta Math. 149 ( 1982 ), 1 - 41 . MR 674165 | Zbl 0547.32012 · Zbl 0547.32012 · doi:10.1007/BF02392348
[4] Z. Blocki , Estimates for the complex Monge-Ampère operator , Bull. Polish Acad. Sci. Math. 41 ( 1993 ), 151 - 157 . Zbl 0795.32003 · Zbl 0795.32003
[5] Z. Błocki , On the Lp-stability for the complex Monge-Ampère operator , Michigan Math. J. 42 ( 1995 ), 269 - 275 . Article | Zbl 0841.35017 · Zbl 0841.35017 · doi:10.1307/mmj/1029005228
[6] Z. Błocki , Smooth exhaustion function in convex domains , to appear in Proc. Amer. Math. Soc. Zbl 0889.35030 · Zbl 0889.35030 · doi:10.1090/S0002-9939-97-03571-5
[7] U. Cegrell , Capacities in complex analysis, Aspect of Math . E14 , Vieweg , 1988 . MR 964469 | Zbl 0655.32001 · Zbl 0655.32001
[8] U. Cegrell - L. Persson , The Dirichlet problem for the complex Monge-Ampère operator: Stability in L2 , Michigan Math. J. 39 ( 1992 ), 145 - 151 . Article | MR 1137895 | Zbl 0799.32013 · Zbl 0799.32013 · doi:10.1307/mmj/1029004461
[9] S.S. Chern - H. Levine - L. Nirenberg , Intrinsic norms on a complex manifold, Global Analysis , Univ. of Tokyo Press , 1969 , 119 - 139 . MR 254877 | Zbl 0202.11603 · Zbl 0202.11603
[10] J.-P. Demailly , Mesures de Monge-Ampère et mesures plurisousharmoniques , Math. Z. 194 ( 1987 ), 519 - 564 . Article | MR 881709 | Zbl 0595.32006 · Zbl 0595.32006 · doi:10.1007/BF01161920
[11] J.-P. Demailly , Potential theory in several complex variables , preprint, 1989 .
[12] J.L. Doob , Classical potential theory and its probabilistic counterpart, Grundl. der Math. Wiss. 262 , Springer-Verlag , 1984 . MR 731258 | Zbl 0549.31001 · Zbl 0549.31001
[13] B. Gaveau , Méthodes de contrôle optimal en analyse complexe I. Résolution d’équations de Monge-Ampère , J. Funct. Anal . 25 ( 1977 ), 391 - 411 . MR 457783 | Zbl 0356.35071 · Zbl 0356.35071 · doi:10.1016/0022-1236(77)90046-5
[14] D. Gilbarg - N.S. Trudinger , Elliptic partial differential equations of second order, Grundl. der Math. Wiss . 244 , Springer Verlag , 1983 . MR 737190 | Zbl 0562.35001 · Zbl 0562.35001
[15] L. Hörmander , An introduction to complex analysis in several variables , North-Holland , 1990 . MR 1045639 | Zbl 0685.32001 · Zbl 0685.32001
[16] N. Kerzman - J.-P. Rosay , Fonctions plurisousharmoniques d’exhaustion bornées et domaines taut , Math. Ann. 257 ( 1981 ), 171 - 184 . MR 634460 | Zbl 0451.32012 · Zbl 0451.32012 · doi:10.1007/BF01458282
[17] M. Klimek , Pluripotential theory , Clarendon Press , 1991 . MR 1150978 | Zbl 0742.31001 · Zbl 0742.31001
[18] S. Kołodziej , Some sufficient condition for solvability of the Dirichlet problem for the complex Monge-Ampère operator , Ann. Polon. Math. 65 ( 1996 ), 11 - 21 . Zbl 0878.32014 · Zbl 0878.32014
[19] N. Levenberg - M. Okada , On the Dirichlet problem for the complex Monge-Ampère operator , Michigan Math. J. 40 ( 1993 ), 507 - 526 . Article | MR 1236176 | Zbl 0805.31004 · Zbl 0805.31004 · doi:10.1307/mmj/1029004835
[20] J. Rauch - B.A. Taylor , The Dirichlet problem for the multidimensional Monge Ampère equation , Rocky Mountain Math. J. 7 ( 1977 ), 345 - 364 . MR 454331 | Zbl 0367.35025 · Zbl 0367.35025 · doi:10.1216/RMJ-1977-7-2-345
[21] R. Richberg , Stetige streng pseudokonvexe Funktionen , Math. Ann. 175 ( 1968 ), 257 - 286 . MR 222334 | Zbl 0153.15401 · Zbl 0153.15401 · doi:10.1007/BF02063212
[22] W. Rudin , Function theory in the unit ball of Cn, Grundl. der Math. Wiss . 241 , Springer-Verlag , 1980 . MR 601594 | Zbl 0495.32001 · Zbl 0495.32001
[23] N. Sibony , Une classe de domaines pseudoconvexes , Duke Math. J. 55 ( 1987 ), 299 - 319 . Article | MR 894582 | Zbl 0622.32016 · Zbl 0622.32016 · doi:10.1215/S0012-7094-87-05516-5
[24] J.B. Walsh , Continuity of envelopes of plurisubharmonic functions , J. Math. Mech. 18 ( 1968 ), 143 - 148 . MR 227465 | Zbl 0159.16002 · Zbl 0159.16002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.