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Continuous solutions of Dirichlet problem to Hessian type equations for $(ω,m)-β$-subharmonic functions on a ball in $\mathbb{C}^n$

Le Mau Hai, Nguyen Van Phu, Trinh Tung

Abstract

In this paper, we investigate the continuity of solutions to the Dirichlet problem for complex Hessian-type equations associated with $(ω, m)-β$-subharmonic functions on a ball in $\mathbb{C}^n$, where $ β=d d^c\|z\|^2=\frac{i}{2} \sum_{j=1}^n d z_j \wedge d \bar{z}_j $ is denoted the flat metric on $\mathbb{C}^n$.

Continuous solutions of Dirichlet problem to Hessian type equations for $(ω,m)-β$-subharmonic functions on a ball in $\mathbb{C}^n$

Abstract

In this paper, we investigate the continuity of solutions to the Dirichlet problem for complex Hessian-type equations associated with -subharmonic functions on a ball in , where is denoted the flat metric on .

Paper Structure

This paper contains 5 sections, 10 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and auxiliary results
  3. Continuous solutions of the Dirichlet problem to Hessian type equation for $(\omega,m)-\beta$-sh functions on a ball in $\mathbb C^n$
  4. Proof of the existence of solutions of the equation \ref{['eq1.5']}
  5. Proof of the continuity of solutions of the equation \ref{['1.7']}

Key Result

Theorem 1.1

Assume that the following conditions are satisfied: (a) $F(t,z)$ is semi-upper continuous on $\mathbb{R}\times\Omega$ and $t\mapsto F(t,z)$ is continuous. (b) There exist a function $0<G\in L^1_{loc}(\Omega,\mu)$ and $v\in SH_{m}^-(\Omega)\cap L^{\infty}(\Omega)$ satisfying $F(t,z)\leq G(z)$ for all Then the following assertions hold: (i) The Dirichlet problem eq1.5 has a solution. Moreover, if $F

Theorems & Definitions (21)

  • Theorem 1.1: Main Theorem
  • Proposition 2.1
  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Theorem 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • ...and 11 more