Complex Hessian-type equations in the weighted \(m\)-subharmonic class. (English) Zbl 1531.32043
Ukr. Math. J. 75, No. 6, 921-935 (2023) and Ukr. Mat. Zh. 75, No. 6, 805-816 (2023).
Summary: We study the existence of a solution to a general type of complex Hessian equation on some Cegrell classes. For a given measure \(\mu\) defined on an \(m\)-hyperconvex domain \(\Omega \subset \mathbb{C}^n\), under suitable conditions, we prove that the equation \(\chi (\cdot) H_m(\cdot) = \mu\) has a solution that belongs to the class \(\mathcal{E}_{ m, \chi } (\Omega)\).
MSC:
| 32U05 | Plurisubharmonic functions and generalizations |
| 32W50 | Other partial differential equations of complex analysis in several variables |
References:
| [1] | H. Amal, S. Asserda, and A. El Gasmi, “Weak solutions to the complex Hessian type equations for arbitrary measures,” Complex Anal. Oper. Theory, 14 (2020). · Zbl 1453.32040 |
| [2] | S. Benelkourchi, V. Guedj, and A. Zeriahi, “Plurisubharmonic functions with weak singularities,” in: Complex Analysis and Digital Geometry: Proc. Kiselmanfest, 2006, Uppsala University, Uppsala (2009), pp. 57-74. · Zbl 1200.32021 |
| [3] | Błocki, Z., Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55, 5, 1735-1756 (2005) · Zbl 1081.32023 · doi:10.5802/aif.2137 |
| [4] | Cegrell, U., Pluricomplex energy, Acta Math., 180, 187-217 (1998) · Zbl 0926.32042 · doi:10.1007/BF02392899 |
| [5] | Cegrell, U., The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier (Grenoble), 54, 159-179 (2004) · Zbl 1065.32020 · doi:10.5802/aif.2014 |
| [6] | D. T. Chuyen, V. T. Thanh, H. T. Lam, and D. T. Duong, “Some relation and comparison principle between the classes \(\mathcal{F}_m\),χ, \( \mathcal{E}_m\), and \(\mathcal{N}_m\),” Tap Chi Khoa Hoc — Dai Hoc Tay Bac, 20, 95-103 (2020). |
| [7] | R. Czyz, “On a Monge-Ampère type equation in the Cegrell class \(\mathcal{E}_χ\),” Ann. Polon. Math., 99, 89-97 (2010). · Zbl 1218.32019 |
| [8] | A. El Gasmi, “The Dirichlet problem for the complex Hessian operator in the class N_m(Ω, f),” Math. Scand., 127, 287-316 (2021). · Zbl 1487.32209 |
| [9] | L. M. Hai, P. H. Hiep, N. X. Hong, and N. V. Phu, “The Monge-Ampére type equation in the weighted pluricomplex energy class,” Internat. J. Math., 25, No. 5, Article 1450042 (2014). · Zbl 1301.32023 |
| [10] | L. M. Hai and V. Van Quan, “Weak solutions to the complex m-Hessian equation on open subsets of \({\mathbb{C}}^n\),” Complex Anal. Oper. Theory, 13, 4007-4025 (2019). · Zbl 1462.32043 |
| [11] | J. Hbil and M. Zaway, Some Results on Complex m-Subharmonic Classes; ArXiv:2201.06851. · Zbl 1522.32087 |
| [12] | Hung, VV, Local property of a class of m-subharmonic functions, Vietnam J. Math., 44, 3, 621-630 (2016) · Zbl 1352.32012 · doi:10.1007/s10013-015-0176-5 |
| [13] | Hung, VV; Phu, NV, Hessian measures on m-polar sets and applications to the complex Hessian equations, Complex Var. Elliptic Equat., 8, 1135-1164 (2017) · Zbl 1371.32018 · doi:10.1080/17476933.2016.1273907 |
| [14] | Lu, CH, A variational approach to complex Hessian equations in C^n, J. Math. Anal. Appl., 431, 1, 228-259 (2015) · Zbl 1326.32056 · doi:10.1016/j.jmaa.2015.05.067 |
| [15] | C. H. Lu, Equations Hessiennes Complexes, Ph. D. Thesis, Univ. Paul Sabatier, Toulouse, France (2012); http://thesesups.ups-tlse.fr/1961/. |
| [16] | Sadullaev, AS; Abdullaev, BI, Potential theory in the class of m-subharmonic functions, Tr. Mat. Inst. Steklova, 279, 166-192 (2012) · Zbl 1298.31010 |
| [17] | N. V. Thien, “Maximal m-subharmonic functions and the Cegrell class \(\mathcal{N}_m\), Indag. Math. (N.S.), 30, 717-739 (2019). · Zbl 1425.32029 |
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.
