Weak Solutions to Complex Monge-Ampère Equations on Compact Kähler Manifold
We show a general existence theorem to the complex Monge-Ampère type equation on compact Kähler manifolds.
math.CV↗
arXiv subjects
Slimane Benelkourchi
Publications and source records attributed to Slimane Benelkourchi.
Weak Solutions To Complex Monge-Ampère Equation on hyperconvex domains
We show a very general existence theorem to the complex Monge-Ampère type equation on hyperconvex domains.
math.CV↗
Weighted Pluricomplex energy II
We continue our study of the Complex Monge-Ampère Operator on the Weighted Pluricomplex energy classes. We give more characterizations of the range of the classes $\mathcal E_ χ$ by the Complex Monge-Ampère Operator. In particular, we prove that a non-negative Borel measure $μ$ is the Monge-Ampère of a unique function $φ\in \mathcal E_χ$ if and only if $χ(\mathcal E_χ) \subset L^1(dμ).$ Then we show that if $μ= (dd^c φ)^n $ for some $φ\in \mathcal E_χ$ then $μ= (dd^c u )^n $ for some $u \in \mathcal E_χ(f) $ where $f$ is a given boundary data. If moreover, the non-negative Borel measure$μ$ is suitably dominated by the Monge-Ampère capacity, we establish a priori estimates on the capacity of sub-level sets of the solutions. As consequence, we give a priori bounds of the solution of the Dirichlet problem in the case when the measure has a density in some Orlicz space.
math.CV↗
Envelope Approach to Degenerate Complex Monge-Ampère Equations on compact Kähler manifold
We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.
math.DG↗
Weighted pluricomplex energy
We study the complex Monge-Ampre operator on the classes of finite pluricomplex energy $\mathcal{E}_χ(Ω)$ in the general case ($χ(0)=0$ i.e. the total Monge-Ampre mass may be infinite). We establish an interpretation of these classes in terms of the speed of decrease of the capacity of sublevel sets and give a complete description of the range of the operator $(dd^c \cdot)^n$ on the classes $\mathcal{E}χ(Ω).$
math.CV↗