scholarly journals Subsolution theorem and the Dirichlet problem for the quaternionic Monge–Ampère equation

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1673-1690 ◽  
Author(s):  
Dongrui Wan
Keyword(s):  
Dirichlet Problem ◽  
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.

scholarly journals On Dirichlet's principle and problem

2012 ◽  
Vol 110 (2) ◽  
pp. 235 ◽  
Author(s):  
Per Åhag ◽  
Urban Cegrell ◽  
Rafal Czyz
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Complete Characterization ◽  
Plurisubharmonic Functions ◽  
Monge Ampere

The aim of this paper is to give a new proof of the complete characterization of measures for which there exists a solution of the Dirichlet problem for the complex Monge-Ampere operator in the set of plurisubharmonic functions with finite pluricomplex energy. The proof uses variational methods.

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