Weak solutions to the complex Monge-Ampere equation on open subsets of $\mathbb{C}^n$

V.\ V.\ Quan; L.\ M.\ Hai

doi:10.15330/ms.51.1.143-151

In the paper, we prove the existence of weak solutions to the complex Monge-Ampere equation in the class $\mathcal{D}(\Omega)$ on an open subset $\Omega$ of $\mathbb C^n$.

1. P. Ahag, U. Cegrell, R. Czyz, P.H. Hiep, Monge-Ampere measures on pluripolar sets, J. Math. Pures Appl., 92 (2009), 613-627.

2. P. Ahag, R. Czyz, Kolodziej's subsolution theorem for unbounded pseudoconvex domains, Universitatis Iagellonicae Acta Mathematica, 50 (2012), 7-23.

3. E. Bedford, B.A. Taylor, The Dirichlet problem for a complex Monge-Ampere operator, Invent. Math., 37 (1976), 1-44.

4. E. Bedford, B.A. Taylor, A new capacity for plurisubharmonic functions, Acta Math., 149 (1982), 1-41.

5. Z. Blocki, On the definition of the Monge-Ampere operator in C2, Math. Ann., 328 (2004), 415-423.

6. Z. Blocki, The domain of definition of the complex Monge-Ampere operator, Amer. J. Math., 128 (2006), 519-530.

7. Z. Blocki, A note on maximal plurisubharmonic functions, Uzbek. Math. J., 1 (2009), 28-32.

8. U. Cegrell, Pluricomplex energy, Acta Math., 180 (1998), 187-217.

9. U. Cegrell, The general definition of the complex Monge-Ampere operator, Ann. Inst. Fourier., 54 (2004), 159-179.

10. U. Cegrell, A general Dirichlet problem for the complex Monge-Ampere operator, Ann. Polon. Math., 94 (2008), 131-147.

11. U. Cegrell, Convergence in capacity, Canad. Math. Bull., 55 (2012), 242-248.

12. L.M. Hai, N.V. Khue, P.H. Hiep, The complex Monge-Ampere operator on bounded domains in $\mathbb{C}^n$, Result. Math., 54 (2009), 309-328.

13. L.M. Hai, N.V. Trao, N.X. Hong, The complex Monge-Ampere equation in unbounded hyperconvex domains in Cn, Complex Variables and Elliptic Equations, 59 (2014), №12, 1758-1774.

14. L.M. Hai, T.V. Thuy, N.X. Hong, A note on maximal subextensions of plurisubharmonic functions, Acta Math Vietnam, 43 (2018), 137-146.

15. N.X. Hong, Monge-Ampere measures of maximal subextensions of plurisubharmonic functions with given boundary values, Complex Var. Elliptic Equ., 60 (2015), №3, 429-435.

16. N.X. Hong, N.V. Trao, T.V. Thuy, Convergence in capacity of plurisubharmonic function functions with given boundary values, Inter. J. Math, 28, №3, (2017) 14 p.

17. P.H. Hiep, Pluripolar sets and the subextension in Cegrell's classes, Complex Var. Elliptic Equ., 53 (2008), №7, 675-684.

18. N.V. Khue, P.H. Hiep, A comparison principle for the complex Monge-Ampere operator in Cegrell's classes and applications, Trans. Am. Math. Soc., 361 (2009), 5539-5554.

19. M. Klimek, Pluripotential Theory, The Clarendon Press Oxford University Press, New York, 1991, Oxford Science Publications.

20. S. Ko lodziej, The range of the complex Monge-Ampere operator, II, Indiana U. Math. J., 44 (1995), 765-782.

21. Y.T. Siu, Extension of meromorphic maps into KЎ§ahler manifolds, Ann. of Math., 102 (1975), №2, 421-462.

	Weak solutions to the complex Monge-Ampere equation on open subsets of $\mathbb{C}^n$
Author	V. V. Quan¹, L. M. Hai² vuquanhau.edu@gmail.com¹, mauhai@fpt.vn² 1) Hanoi Architectural University, Hanoi, Viet Nam; 2) Department of Mathematics, Hanoi National University of Education Xuan Thuy Street, Cau Giay, Hanoi, Viet Nam
Abstract	In the paper, we prove the existence of weak solutions to the complex Monge-Ampere equation in the class $\mathcal{D}(\Omega)$ on an open subset $\Omega$ of $\mathbb C^n$.
Keywords	Monge-Ampere equation; weak solution; plurisubharmonic function; hyperconvex domain
DOI	doi:10.15330/ms.51.2.143-151
Reference	1. P. Ahag, U. Cegrell, R. Czyz, P.H. Hiep, Monge-Ampere measures on pluripolar sets, J. Math. Pures Appl., 92 (2009), 613-627. 2. P. Ahag, R. Czyz, Kolodziej's subsolution theorem for unbounded pseudoconvex domains, Universitatis Iagellonicae Acta Mathematica, 50 (2012), 7-23. 3. E. Bedford, B.A. Taylor, The Dirichlet problem for a complex Monge-Ampere operator, Invent. Math., 37 (1976), 1-44. 4. E. Bedford, B.A. Taylor, A new capacity for plurisubharmonic functions, Acta Math., 149 (1982), 1-41. 5. Z. Blocki, On the definition of the Monge-Ampere operator in C2, Math. Ann., 328 (2004), 415-423. 6. Z. Blocki, The domain of definition of the complex Monge-Ampere operator, Amer. J. Math., 128 (2006), 519-530. 7. Z. Blocki, A note on maximal plurisubharmonic functions, Uzbek. Math. J., 1 (2009), 28-32. 8. U. Cegrell, Pluricomplex energy, Acta Math., 180 (1998), 187-217. 9. U. Cegrell, The general definition of the complex Monge-Ampere operator, Ann. Inst. Fourier., 54 (2004), 159-179. 10. U. Cegrell, A general Dirichlet problem for the complex Monge-Ampere operator, Ann. Polon. Math., 94 (2008), 131-147. 11. U. Cegrell, Convergence in capacity, Canad. Math. Bull., 55 (2012), 242-248. 12. L.M. Hai, N.V. Khue, P.H. Hiep, The complex Monge-Ampere operator on bounded domains in $\mathbb{C}^n$, Result. Math., 54 (2009), 309-328. 13. L.M. Hai, N.V. Trao, N.X. Hong, The complex Monge-Ampere equation in unbounded hyperconvex domains in Cn, Complex Variables and Elliptic Equations, 59 (2014), №12, 1758-1774. 14. L.M. Hai, T.V. Thuy, N.X. Hong, A note on maximal subextensions of plurisubharmonic functions, Acta Math Vietnam, 43 (2018), 137-146. 15. N.X. Hong, Monge-Ampere measures of maximal subextensions of plurisubharmonic functions with given boundary values, Complex Var. Elliptic Equ., 60 (2015), №3, 429-435. 16. N.X. Hong, N.V. Trao, T.V. Thuy, Convergence in capacity of plurisubharmonic function functions with given boundary values, Inter. J. Math, 28, №3, (2017) 14 p. 17. P.H. Hiep, Pluripolar sets and the subextension in Cegrell's classes, Complex Var. Elliptic Equ., 53 (2008), №7, 675-684. 18. N.V. Khue, P.H. Hiep, A comparison principle for the complex Monge-Ampere operator in Cegrell's classes and applications, Trans. Am. Math. Soc., 361 (2009), 5539-5554. 19. M. Klimek, Pluripotential Theory, The Clarendon Press Oxford University Press, New York, 1991, Oxford Science Publications. 20. S. Ko lodziej, The range of the complex Monge-Ampere operator, II, Indiana U. Math. J., 44 (1995), 765-782. 21. Y.T. Siu, Extension of meromorphic maps into KЎ§ahler manifolds, Ann. of Math., 102 (1975), №2, 421-462.
Pages	143-151
Volume	51
Issue	2
Year	2019
Journal	Matematychni Studii
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