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A Dirichlet type problem for non-pluripolar complex Monge-Ampère equations

Thai Duong Do, Hoang-Son Do, Van Tu Le, Ngoc Thanh Cong Pham

TL;DR

This work studies a Dirichlet-type problem for the non-pluripolar Monge-Ampère operator $(NP(dd^c u))^n$ with prescribed singularity $P[u]=\phi$ on bounded domains in $\mathbb{C}^n$, allowing non-hyperconvex geometry and model-type singularities. It develops an envelope method together with new Xing-type comparison principles to construct solutions and establish uniqueness under mild non-pluripolar hypotheses, notably when there exists $v$ with $(NP(dd^c v))^n\ge\mu$ and $P[v]=\phi$. The main result shows that the supremum envelope $u_S=(\sup\{w: w\le\phi,\; NP(dd^c w)^n\ge\mu\})^*$ solves $(NP(dd^c u))^n=\mu$ with $P[u]=\phi$, with uniqueness in the presence of a barrier $\psi\in\mathcal{N}_{NP}$ satisfying $NP(dd^c\psi)^n\ge\mu$. The paper extends prior work of Darvas–Di Nezza–Lu and Ahag–Cegrell–Czyż–Pham by providing a local solvability framework that does not require the domain to be hyperconvex or $\phi$ to lie in a restrictive class, and delivers tools applicable to non-pluripolar measures and model-type singularities.

Abstract

In this paper, we study a Dirichlet type problem for the non-pluripolar complex Monge - Ampère equation with prescribed singularity on a bounded domain of $\mathbb{C}^n$. We provide a local version for an existence and uniqueness theorem proved by Darvas, Di Nezza and Lu. Our work also extends a result of Ahag, Cegrell, Czyz and Pham.

A Dirichlet type problem for non-pluripolar complex Monge-Ampère equations

TL;DR

This work studies a Dirichlet-type problem for the non-pluripolar Monge-Ampère operator with prescribed singularity on bounded domains in , allowing non-hyperconvex geometry and model-type singularities. It develops an envelope method together with new Xing-type comparison principles to construct solutions and establish uniqueness under mild non-pluripolar hypotheses, notably when there exists with and . The main result shows that the supremum envelope solves with , with uniqueness in the presence of a barrier satisfying . The paper extends prior work of Darvas–Di Nezza–Lu and Ahag–Cegrell–Czyż–Pham by providing a local solvability framework that does not require the domain to be hyperconvex or to lie in a restrictive class, and delivers tools applicable to non-pluripolar measures and model-type singularities.

Abstract

In this paper, we study a Dirichlet type problem for the non-pluripolar complex Monge - Ampère equation with prescribed singularity on a bounded domain of . We provide a local version for an existence and uniqueness theorem proved by Darvas, Di Nezza and Lu. Our work also extends a result of Ahag, Cegrell, Czyz and Pham.
Paper Structure (10 sections, 25 theorems, 168 equations)

This paper contains 10 sections, 25 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The plurifine topology
  4. The non-pluripolar complex Monge-Ampère measure
  5. Stability of subsolutions and supersolutions
  6. An envelope of plurisubharmonic functions
  7. Xing-type comparison principles
  8. Proofs of Theorem \ref{['main']} and Corollary \ref{['cormain']}
  9. Proof of Theorem \ref{['main']}
  10. Proof of Corollary \ref{['cormain']}

Key Result

Theorem 1.2

Assume that there exists $v\in\text{PSH}^-(\Omega)$ such that $\textnormal{NP}({dd^cv})^n\geq\mu$ and $P[v]=\phi$. Denote Then $u_S:=(\sup\{w: w\in S\})^*$ is a solution of the problem NPMA. Moreover, if there exists $\psi\in\mathcal{N}_\textnormal{NP}$ such that $\textnormal{NP}({dd^c\psi})^n\geq\mu$ then $u_S$ is the unique solution of NPMA.

Theorems & Definitions (48)

  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 38 more