Báo cáo toán học: "A Remark on the Dirichlet Problem"

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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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9LHWQDP -RXUQDO Vietnam 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, 2005 Abstract. Given a positive measure μ on a strongly pseudoconvex domain in Cn . We study 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 class P SH (Ω) ∩ L∞ (Ω) of locally bounded psh functions on Ω, more precisely, if loc u ∈ P SH (Ω) ∩ L∞ (Ω) is a positive Borel measure. Furthermore, this operator loc is 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 class of psh functions which are locally bounded near ∂ Ω. Recently in [5, 6], Cegrell introduced 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 these papers, he also studied the Dirichlet problems for the classes Fp (see Sec. 2 for details). The aim of our work is to investigate the Dirichlet problem for a new class of psh function. This class consist, in particular, the sum of a function in a the 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, Vietnam
336 Pham Hoang Hiep Main 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 psh functions which are bounded near ∂ Ω was proved by Xing in [13] and for the classes 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. Preliminaries In this section we recall some elements and results of pluripotential theory that will 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 Cn meaning 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 Bedford and 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 E for every Borel set E in Ω. 2.2. According to Xing (see [13]), a sequence of positive measures {μj } on Ω is called 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 a 2.3. By Bloc = Bloc (Ω) we denote the set of upper bounded psh functions u which 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, (ddc ϕ)n < +∞ , z →∂ Ω Ω (−ϕj )p (ddc ϕj )n < +∞ , Ep = Ep (Ω) = ϕ ∈ P SH (Ω) : ∃E0 ϕj ϕ, sup j ≥1 Ω
A Remark on the Dirichlet Problem 337 (−ϕj )p (ddc ϕj )n , Fp = Fp (Ω) = ϕ ∈ P SH (Ω) : ∃E0 ϕj ϕ, sup j ≥1 Ω (ddc ϕj )n < +∞ < + ∞, sup j ≥1 Ω E = E (Ω) = ϕ ∈ P SH (Ω) : ∀z0 ∈ Ω ∃ a neighborhood ω z 0 , E0 ϕj c n ϕ on ω, sup (dd ϕj ) < +∞}. j ≥1 Ω The following inclusions are obvious E0 ⊂ Fp ⊂ Ep ⊂ E . It is also known that these inclusion are strict (see [5, 6]). The interesting theorem below was proved by Cegrell (see [6]) Theorem 2.5. The class E has the following properties 1. E is a convex cone. 2. If u ∈ E , v ∈ P SH − (Ω) = {ϕ ∈ P SH (Ω) : ϕ ≤ 0}, then max(u, v ) ∈ E . 3. If u ∈ E , P SH (Ω) ∩ L∞ (Ω) uj u, then (ddc uj )n is weakly convergent. loc − Conversely if K ⊂ P SH (Ω) satisﬁes 2 and 3, then K ⊂ E − − Since Bloc = Bloc ∩ P SH − (Ω) satisﬁes 2 and 3 we have by [8] Bloc ⊂ E . a 2.6. Cegrell also studied the following Dirichlet problem: Given μ a positive measure on Ω, ﬁnd u ∈ Fp such that (ddc u)n = μ. He gave a necessary and suﬃcient condition for this problem to have a solution (Theorem 5.2 in [5]). 2.7. We deﬁne a a Ep + Bloc = u ∈ P SH (Ω) : ∃ ϕ ∈ Ep , f ∈ Bloc : ϕ + f ≤ u ≤ sup u < +∞ , Ω . a a Fp + Bloc = u ∈ P SH (Ω) : ∃ ϕ ∈ Fp , f ∈ Bloc : ϕ + f ≤ u ≤ sup u < +∞ Ω a It follows that if ϕ + f ≤ u < sup u < +∞, ϕ ∈ E , f ∈ Bloc then Ω u − c = max(u − c, ϕ + f − c) ∈ E , because ϕ + (f − c) ∈ E , where c = max(sup f, sup u). Ω Ω Thus we can deﬁne (ddc u)n for u ∈ Ep + Bloc . a 2.8. The aim of this work is to study a Dirichlet problem similar to the one a a considered by Cegrell but for the classes Ep + Bloc and Fp + Bloc . Namely, given a a a positive measure μ on Ω and h ∈ C (∂ Ω), ﬁnd u ∈ Ep + Bloc (resp. Fp + Bloc ) cn such that (dd u) = μ and lim u(z ) = h(ξ ) ∀ξ ∈ ∂ Ω. z →ξ 2.9. Let μ be a positive measure on Ω and h ∈ C (∂ Ω). Following Cegrell, we deﬁne
338 Pham Hoang Hiep B (μ, h) = {v ∈ P SH (Ω) ∩ L∞ (Ω) : (ddc v )n ≥ μ, lim v (z ) ≤ h(ξ )}, loc z →ξ U (μ, h)(z ) = sup{v (z ) : v ∈ B (μ, h)}, z ∈ Ω. Observe that B (μ, h) = ∅ implies that μ vanishes on pluripolar sets. The function U (μ, h) plays a crucial role in solving the Dirichlet problem. a 3. The Comparison Principle for Ep + Bloc In order to prove the main theorem, in this section we prove the following comparison principle a Theorem 3.1. Let u, v be functions in Ep + Bloc satisfying lim [u(z ) − v (z )] ≥ 0. z →∂ Ω Then (ddc v )n ≤ (ddc u)n . {u −t} an open neighborhood j j j of ∂Dkj (s). It follows that akj (s) ≥ t−n (ddc max(uk , −t))n = t−n (ddc uk )n = t−n bkj (s). j j Dkj (s) Dkj (s)
A Remark on the Dirichlet Problem 339 Letting t s we get sn akj (s) ≥ bkj (s) for k, j ≥ 1 and s > 0. (1) Given > 0. By the hypothesis there exists s0 > 0 such that sn a(s0 ) < . (2) 0 Let E ⊂ D with Cn (E ) < sn . Take an open neighborhood G of E such that 0 c kn (ddc uj )n weakly as k → ∞ we have (dd uj ) → Cn (G) < sn . Since 0 (ddc uj )n ≤ (ddc uj )n ≤ lim (ddc uk )n j k→∞ E G G (ddc uk )n + (ddc uk )n ] ≤ lim [ j j k→∞ Dkj (s0 ) G\Dkj (s0 ) [sn akj (s0 ) sn Cn (G)] ≤ sn a(s0 ) + < 2 ≤ lim + 0 0 0 k→∞ for j ≥ 1. Hence (ddc uj )n Cn in D uniformly for j ≥ 1. Proof of Theorem 3.1. We may assume that u, v ≤ 0 and lim [u(z ) − v (z )] > z →∂ Ω − a δ > 0. By hypothesis u, v ∈ Ep + Bloc it is easy to ﬁnd ϕ ∈ Ep , g ∈ Bloc such that ϕ + g ≤ min(u, v ). Let ϕj ϕ be a sequence decreasing to ϕ as in the deﬁnition of Ep . For each j ≥ 1 put gj = max(g, −j ), uj = max(u, ϕj + gj ), vj = max(v, ϕj + gj ). It follows that gj , uj , vj are bounded and gj g , uj u, vj v . By the comparison principle for bounded psh functions we have (ddc vk )n ≤ (ddc uj )n {uj
340 Pham Hoang Hiep (ddc v )n ≤ (ddc u)n . {u 0 such that z →ξ0 ω (z ) < − for z ∈ B (ξ0 , δ ) ∩ Ω. Let τ ∈ C (∂ Ω) such that τ |B (ξ0 , δ )∩∂ Ω = , 2 suppτ ⊂ B (ξ0 , δ ) ∩ ∂ Ω. By [2] there exists φ ∈ P SH (Ω) ∩ C (Ω) such that ¯ (ddc φ)n = 0 and φ|∂ Ω = τ . Since lim [ωk (z ) − (ω (z ) + φ(z ))] ≥ 0 for ξ ∈ ∂ Ω z →ξ and (ddc ωk )n = μk ≤ μ = (ddc ω )n ≤ (ddc (ω + φ))n , we have ωk ≥ ω + φ on Ω for k ≥ 1. Thus ω ≥ ω + φ on Ω. Hence φ ≤ 0 on Ω\{ω = −∞}. Since φ is plurisubharmonic, φ ≤ 0 on Ω. This is impossible, because φ(ξ ) = τ (ξ ) = for δ ξ ∈ B (ξ0 , 2 ) ∩ ∂ Ω. Hence lim ω (z ) = 0 for ξ ∈ ∂ Ω. From the relations z →ξ U ((ddc (ωk + U (0, h)))n , h) = ωk + U (0, h), (ddc (ω + U (0, h)))n ≥ μk , and from Theorem 8.1 in [5] it follows that (ddc U (μk , h))n = μk , U (0, h) ≥ U (μk , h) ≥ ωk + U (0, h). u ∈ Ep + Bloc with (ddc u)n = μ and a Theorem 3.1 implies that U (μk , h) U (0, h) ≥ u ≥ ω + U (0, h). Thus for ξ ∈ ∂ Ω we have h(ξ ) = lim U (0, h) ≥ lim u(z ) ≥ lim [ω (z ) + U (0, h)(z )] z →ξ z →ξ z →ξ = lim ω (z ) + lim U (0, h)(z ) = h(ξ ). z →ξ z →ξ Consequently u ∈ Ep + Bloc such that (ddc u)n = μ and lim u(z ) = h(ξ ) ∀ξ ∈ a z →ξ ∂ Ω.
A Remark on the Dirichlet Problem 341 (ii) Let {Ωj } be an increasing exhaustion sequence of strongly pseudoconvex subdomains of Ω. For each j ≥ 1 take a sequence of distinguished points zjm ⊂ Ωj \Ωj −1 converging to ξj ∈ ∂ Ωj as m → ∞ and a sequence sj 0 such that B (zjm , sjm )⊂ Ωj \Ωj −1 and B (zjm , sjm ) ∩ B (zjt , sjt ) = ∅ for m = t. Let ∞ ajm < ∞. Put ajm > 0 with j,m=1 ajm f= 2n χB (zjm ,rjm ) , dn rjm j,m≥1 where 0 < rjm < sjm are chosen such that 1 p (Cn (B (zjm , rjm ), Ω)) n+p → 0 as m → ∞, ajm for j ≥ 1 and dn is the volume of the unit ball in Cn . Assume that f dλ ≤ (ddc u)n for some u ∈ Ep + Bloc . Take ϕ ∈ Ep , g ∈ Bloc a a such that ϕ + g ≤ u ≤ sup u < +∞. We may assume that g and u are negative. Ω Let j0 ≥ 2 and M > 0 such that g > −M on Ωj0 \Ωj0 −1 . Put M g = max(g, AhΩj0 ) where A = − ˜ > 0. sup hΩj0 ¯ Ωj0 It follows that g ∈ E0 , g = g on Ωj0 \Ωj0 −1 . ˜ ˜ Let u = max(u, ϕ + g). Since ϕ + g ≤ u ≤ 0 and ϕ + g ∈ Ep + E0 = Ep , by ˜ ˜ ˜˜ ˜ [5] we have u ∈ Ep . ˜ Moreover u = u on Ωj0 \Ωj0 −1 . Thus for Bm = B (zj0 m , rj0 m ) we have ˜ (ddc u)n = (ddc u)n . aj0 m = f dλ = ˜ Bm Bm Bm u as in the deﬁnition of Ep . Then (ddc uk )n → (ddc u)n weakly (see Let uk ˜ ˜ ˜ ˜ [5]). Applying the Holder inequality (see [7]) we have (ddc u)n ≤ lim (ddc uk )n aj0 m = ˜ ˜ k→∞ Bm Bm (−hBm )p (ddc uk )n = lim ˜ k→∞ Bm p n ≤ α1 lim [ (−hBm )p (ddc hBm )n ] n+p [ (−uk )p (ddc uk )n ] n+p ˜ ˜ k→∞ Ω Ω p c n ≤ α2 [ (dd hBm ) ] n +p Ω p = α2 [Cn (Bm , Ω)] n+p , n where α2 = α1 [sup (−uk )p (ddc uk )n ] n+p < +∞. This is impossible, because ˜ ˜ k ≥1 Ω
342 Pham Hoang Hiep p [Cn (Bm , Ω)] p+n lim = 0. m→∞ aj0 m Remark. Using Theorem 7.5 in [1] we can ﬁnd u ∈ F a such that (ddc u)n = f dλ where f is constructed as in (ii). Hence, there exists a function u in F a \ (ξp + Bloc ). a Acknowledgements. The author is grateful to Professor Nguyen Van Khue for sug- gesting the problem and for many helpful discussion during the preparation of this work. References 1. P. ˚hag, The Complex Monge–Ampere Operator on Bounded Hyperconvex Do- A mains, Ph. D. Thesis, Ume ˚ University, 2002. a 2. E. Bedford and B. A. Taylor, The Dirichlet problem for the complex Monge- Ampere operator, Invent. Math. 37 (1976) 1–44. 3. E. Bedford and B. A.Taylor, A new capacity for plurisubharmonic function, Acta Math. 149 (1982) 1–40. 4. Z. Blocki, The Complex Monge-Ampere Operator in Pluripotential Theory, Lec- ture Notes, 1998. 5. U. Cegrell, Pluricomplex energy, Acta Math. 180 (1998) 187–217. 6. U. Cegrell, The general deﬁnition of the complex Monge-Ampere operator, Ann. Inst. Fourier (Grenoble) 54 (2004) 159–179. 7. U. Cegrell and L. Persson, An energy estimate for the complex Monge-Ampere operator, Annales Polonici Mathematici (1997) 96–102. 8. U. Cegrell, S. Kolodziej, and A.Zeriahi, Subextension of plurisubharmonic func- tions with weak singularities, Math. Zeit. (to appear). 9. J - P. Demailly, Monge-Ampere operators, Lelong Numbers and Intersection the- ory, Complex Analysis and Geometry, Univ. Ser. Math., Plenum, New York, 1993, 115–193. 10. P. H. Hiep, A characterization of bounded plurisubharmonic functions, Annales Polonici Math. 85(2005) 233–238. 11. N. V. Khue, P. H. Hiep, Complex Monge-Ampere measures of plurisubharmonic functions which are locally bounded near the boundary, Preprint 2004. 12. M. Klimek, Pluripotential Theory, Oxford, 1990. 13. Y. Xing, Complex Monge-Ampere measures of pluriharmonic functions with bounded values near the boundary, Cand. J. Math. 52 (2000) 1085–1100.