scholarly journals A correction to “The Dirichlet problem for complex Monge–Ampère equations and regularity of the pluri-complex Green function”

2000 ◽  
Vol 8 (1) ◽  
pp. 213-218 ◽  
Author(s):  
Bo Guan
Keyword(s):  
Dirichlet Problem ◽  
Green Function ◽  
Complex Green Function ◽  
Monge Ampere

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

2018 ◽  
Vol 18 (2) ◽  
pp. 289-302
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Crucial Role ◽  
Blow Up ◽  
Boundary Behavior ◽  
Structure Condition ◽  
Bounded Smooth Domain ◽  
Convex Solution ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

AbstractThis paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation\operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0,where Ω is a strictly convex and bounded smooth domain in{\mathbb{R}^{N}}, with{N\geq 2},{g\in C^{1}((0,\infty),(0,\infty))}is decreasing in{(0,\infty)}and satisfies{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and{b\in C^{\infty}(\Omega)}is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition ongwhich plays a crucial role in the boundary behavior of such solution.

scholarly journals A Remark on the Continuous Subsolution Problem for the Complex Monge-Ampère Equation

2019 ◽  
Vol 45 (1) ◽  
pp. 83-91
Author(s):  
Sławomir Kołodziej ◽  
Ngoc Cuong Nguyen
Keyword(s):  
Dirichlet Problem ◽  
Modulus Of Continuity ◽  
Continuous Solution ◽  
Type Condition ◽  
Right Hand ◽  
The Right ◽  
Monge Ampere

Abstract We prove that if the modulus of continuity of a plurisubharmonic subsolution satisfies a Dini-type condition then the Dirichlet problem for the complex Monge-Ampère equation has the continuous solution. The modulus of continuity of the solution also given if the right hand side is locally dominated by capacity.

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