Báo cáo toán học: A Remark on the Dirichlet Problem

Cho một μ biện pháp tích cực trên một miền mạnh mẽ pseudoconvex Cn. Chúng tôi nghiên cứu vấn đề Dirichlet (DDC u) n = μ trong một lớp mới của chức năng plurisubharmonic. Lớp này bao gồm Ep các lớp học (p ≥ 1) được giới thiệu bởi Cegrell trong [5].1.
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Báo cáo toán học: "A Remark on the Dirichlet Problem" 9LHWQDP -RXUQDOVietnam Journal of Mathematics 33:3 (2005) 335–342 RI 0$7+(0$7,&6 ‹ 9$67 A Remark on the Dirichlet Problem* Pham Hoang Hiep Department of Mathematics, Hanoi University of Education, 136 Xuan Thuy Street, Cau Giay, Hanoi, Vietnam Received October 06, 2004 Revised March 03, 2005Abstract. Given a positive measure μ on a strongly pseudoconvex domain in Cn . Westudy the Dirichlet problem (ddc u)n = μ in a new class of plurisubharmonic function.This class includes the classes Ep (p ≥ 1) introduced by Cegrell in [5].1. Introduction.Let Ω be a bounded domain in Cn . By P SH (Ω) we denote the set of plurisub-harmonic (psh) functions on Ω. By the fundamental work of Bedford and Taylor[1, 2], the complex Monge-Ampere operator (ddc )n is well defined over the classP SH (Ω) ∩ L∞ (Ω) of locally bounded psh functions on Ω, more precisely, if locu ∈ P SH (Ω) ∩ L∞ (Ω) is a positive Borel measure. Furthermore, this operator locis continuous with respect to increasing and decreasing sequences. Later, De-mailly has extended the domain of definition of the operator (ddc u)n to the classof psh functions which are locally bounded near ∂ Ω. Recently in [5, 6], Cegrellintroduced the largest class of upper bounded psh functions on a bounded hyper-convex domain Ω such that the operator (ddc u)n can be defined on it. In thesepapers, he also studied the Dirichlet problems for the classes Fp (see Sec. 2 fordetails). The aim of our work is to investigate the Dirichlet problem for a newclass of psh function. This class consist, in particular, the sum of a function in athe class Ep and a function in Bloc (see Sec. 2 for the definitions of these classes). Now we are able to formulate the main result of our work∗ This work was supported by the National Research Program for Natural Science, Vietnam336 Pham Hoang HiepMain theorem.(i) Let Ω be a bounded strongly pseudoconvex domain in Cn and let μ be a a positive measure on Ω, h ∈ C (∂ Ω) such that there exists v ∈ Ep + Bloc a ) with (ddc v )n ≥ μ (resp. Fp + Bloc Then there exists u ∈ Ep + Bloc (resp. Fp + Bloc ) such that (ddc u)n = μ a a and lim u(z ) = h(ξ ), ∀ξ ∈ ∂ Ω. z →ξ(ii) There exists f ∈ L1 (Ω) such that there exists no function u ∈ Ep + Bloc a cn which satisfying f dλ ≤ (dd u) . a a For the definitions of Ep + Bloc and Fp + Bloc see Sec. 2. a Note that the main theorem for the subclass B of Bloc consisting of pshfunctions which are bounded near ∂ Ω was proved by Xing in [13] and for theclasses Ep and Fp , p ≥ 1 by Cegrell in [5]. The key element in the proof of our main theorem is a comparison principle(Theorem 3.1), which is an extension of Lemma 4.4, Theorem 4.5 in [5].2. PreliminariesIn this section we recall some elements and results of pluripotential theory thatwill be used through out the paper. All this can be found in [2, 3, 5, 6, 11...].2.0. Unless otherwise specified, Ω will be a bounded hyperconvex domain in Cnmeaning that there exists a negative exhaustive psh function for Ω .2.1. Let Ω be a bounded domain in Cn . The Cn -capacity in the sense of Bedfordand Taylor on Ω is the set function given by (ddc u)n : u ∈ P SH (Ω), −1 ≤ u ≤ 0 Cn (E ) = Cn (E, Ω) = sup Efor every Borel set E in Ω.2.2. According to Xing (see [13]), a sequence of positive measures {μj } on Ω iscalled uniformly absolutely continuous with respect to Cn in a subset E of Ω if ∀ > 0, ∃δ > 0 : F ⊂ E, Cn (F ) < δ ⇒ μj (F ) < , ∀j ≥ 1 Cn in E uniformly for j ≥ 1.We write μj a a2.3. By Bloc = Bloc (Ω) we denote the set of upper bounded psh functions uwhich are locally bounded near ∂ Ω such that (ddc u)n Cn in every E ⊂⊂ Ω.2.4. The following classes of psh functions were introduced by Cegrell in [5]and [6] E0 = E0 (Ω) = ϕ ∈ P SH (Ω) ∩ L∞ (Ω) : lim ϕ(z ) = 0, (dd ...