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Asympotitcs for Some Singular Monge-Ampère Equations

Nicholas McCleerey

Abstract

Given a psh function $\varphi\in\mathcal{E}(Ω)$ and a smooth, bounded $θ\geq 0$, it is known that one can solve the Monge-Ampère equation $\mathrm{MA}(\varphi_θ)=θ^n\mathrm{MA}(\varphi)$, with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czyż--Hiep. Under some natural conditions, we show that $\varphi_θ$ is comparable to $θ\varphi$ on much of $Ω$; especially, it is bounded on the interior of $\{θ= 0\}$. Our results also apply to complex Hessian equations, and can be used to produce interesting Green's functions.

Asympotitcs for Some Singular Monge-Ampère Equations

Abstract

Given a psh function and a smooth, bounded , it is known that one can solve the Monge-Ampère equation , with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czyż--Hiep. Under some natural conditions, we show that is comparable to on much of ; especially, it is bounded on the interior of . Our results also apply to complex Hessian equations, and can be used to produce interesting Green's functions.

Paper Structure

This paper contains 11 sections, 13 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Basics and Notation
  4. Supersolutions and Polar Mass
  5. Extensions
  6. Three-Circle's Theorem and Corollaries
  7. Main Results
  8. Subsolution
  9. Supersolution
  10. Proof of the Theorem
  11. Proof of the Corollary

Key Result

Theorem 1.1

Suppose that $\psi\in \mathcal{E}_m(\Omega)$, the Cegrell class on $\Omega$, is such that $\psi\leqslant -1$ and $\mathrm{H}^m(\psi)$ is supported on $S_\psi := \{\psi = -\infty\}$; moreover, assume that $S_\psi$ is closed and that $\psi$ is smooth on $\Omega\setminus S_\psi$. Additionally, suppose Then for any $\theta\in C^2(\Omega)$, $0\leqslant \theta\leqslant 1$, the envelope: satisfies: Mo

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 19 more