In this article we solve the complex Monge-Ampère equation for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Ko lodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge-Ampère measure, then it is a complex Monge-Ampère measure.2000 Mathematics Subject Classification. Primary 32W20; Secondary 32U15.
We study the geometry of m-regular domains within the CaffarelliNirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every mhyperconvex domain admits an exhaustion function that is negative, smooth, strictly m-subharmonic, and has bounded m-Hessian measure.
Abstract. The aim of this paper is to give a new proof of the complete characterization of measures for which there exist a solution of the Dirichlet problem for the complex Monge-Ampère operator in the set of plurisubharmonic functions with finite pluricomplex energy. The proof uses variational methods. IntroductionThroughout this note let Ω ⊆ C n , n ≥ 1, be a bounded, connected, open, and hyperconvex set. By E 0 we denote the family of all bounded plurisubharmonic functions ϕ defined on Ω such that lim z→ξ ϕ(z) = 0 for every ξ ∈ ∂Ω , andwhere (dd c · ) n is the complex Monge-Ampère operator. Next let E p , p > 0, denote the family of plurisubharmonic functions u defined on Ω such that there exists a decreasing sequence {u j }, u j ∈ E 0 , that converges pointwise to u on Ω, as j tends to +∞, and 10,14]). It should be noted that it follows from [10] that the complex Monge-Ampère operator is well-defined on E p . It is not only within pluripotential theory these cones have been proven useful, but also as a tool in dynamical systems and algebraic geometry (see e.g. [2,17]). For further information on pluripotential theory we refer to [16,19,20]. The purpose of this paper is to give a new proof of Theorem B below and use Theorem B to prove (2) implies (1) the following theorem:Theorem A (Dirichlet's problem). Let µ be a non-negative Radon measure, then the following conditions are equivalent:(1) there exists a function u ∈ E 1 such that (dd c u) n = µ, (2) there exists a constant B > 0, such that Ω (−ϕ) dµ ≤ B e 1 (ϕ) 1 n+1 for all ϕ ∈ E 1 , (1.1) (3) the class E 1 is contained in L 1 (µ),2000 Mathematics Subject Classification. Primary 35J20; Secondary 32W20.
The complex Monge-Ampère operator is a useful tool not only within pluripotential theory, but also in algebraic geometry, dynamical systems and Kähler geometry. In this self-contained survey we present a unified theory of Cegrell's framework for the complex Monge-Ampère operator. Acknowledgements. The author would like to thank PerÅhag and S lawomir Ko lodziej for their generous help and encouragement while writing this survey. Their valuable comments and suggestions essentially improved the final version of this survey.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.