The Bremermann-Dirichlet problem for unbounded domains of $C^n$

无界域上 $q$-多次调和函数的 Perron-Bremermann 包络

The Perron-Bremermann envelope for $q$-plurisubharmonic functions on unbounded domains in $\mathbb{C}^n$

研究无界域上$q$-多次调和函数的Perron-Bremermann包络，证明在边界有$r$-峰值点且有界类型的条件下，有界连续边界函数可延拓为极大有界连续函数，满足$q$-多次调和与$(n-q-1)$-多次超调和性质。

Continuity of envelopes of unbounded plurisubharmonic functions

On bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function $ϕ$ such that $ϕ^*=ϕ_*$ on the closure of the domain. For $ϕ$ satisfying certain additional criteria limiting its behavior at the singularities, we establish a set where the Perron-Bremermann envelope $P ϕ$ is guaranteed to be continuous. This result is a generalization of a classic result in pluripotential theory due to J. B. Walsh. As an application, we show that the complex Monge--Ampère equation \[ (dd^cu)^n = μ\] being uniquely solvable for continuous boundary data implies that …

Kiselman's principle, the Dirichlet problem for the Monge-Ampere equation, and rooftop obstacle problems

First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data. This yields a new relation between solutions of the Dirichlet problem for the homogeneous real and complex Monge-Ampere equations and Kiselman's minimum principle. More generally, it establishes partial regularity for a Bremermann envelope whether or not it solves the Monge-Ampere equation. Second, we prove the second order regularity of the solution of the fre…

具有无界边界值的复Dirichlet问题

Complex Dirichlet problem with unbounded boundary values

研究无界边界数据的复Dirichlet问题，提出基于局部双全纯形变与障碍函数的内部连续性定理，并证明非单调收敛、稳定性与唯一性。

$\mathbb{C}^n$ 中有界拟 B-正则域上多重次调和函数的 Dirichlet 问题

Dirichlet problem for plurisubharmonic functions on bounded quasi B-regular domains in $\mathbb{C}^n$

本文研究了有界域上具有非连续非有界边界数据的极大多重次调和函数的存在性与唯一性，通过探讨 Perron-Bremermann 包络的连续性，实现了对可能上方无界的多重次调和函数的连续逼近。

Level sets of potential functions bisecting unbounded quadrilaterals

We study the mixed Dirichlet-Neumann problem for the Laplace equation in the complement of a bounded convex polygonal quadrilateral in the extended complex plane. The Dirichlet\,/\,Neumann conditions at opposite pairs of sides are $\{0,1\}$ and $\{0,0\},$ resp. The solution to this problem is a harmonic function in the unbounded complement of the polygon known as the \emph{potential function} of the quadrilateral. We compute the values of the potential function including its value at infinity.