Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform $L^\infty$ estimates: subsolutions to fully nonlinear partial differential equations

Bin Guo, Duong H. Phong

Abstract

Uniform bounds are obtained using the auxiliary Monge-Ampère equation method for solutions of very general classes of fully non-linear partial differential equations, assuming the existence of a ${C}$-subsolution in the sense of G. Székelyhidi and B. Guan. The main advantage over previous estimates is in the Kähler case, where the new estimates remain uniform as the background metric is allowed to degenerate.

Uniform $L^\infty$ estimates: subsolutions to fully nonlinear partial differential equations

Abstract

Uniform bounds are obtained using the auxiliary Monge-Ampère equation method for solutions of very general classes of fully non-linear partial differential equations, assuming the existence of a -subsolution in the sense of G. Székelyhidi and B. Guan. The main advantage over previous estimates is in the Kähler case, where the new estimates remain uniform as the background metric is allowed to degenerate.
Paper Structure (7 sections, 8 theorems, 89 equations)

This paper contains 7 sections, 8 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. The Kähler case
  3. The Hermitian case
  4. Other forms of subsolutions
  5. Examples
  6. The $J$-equation
  7. Hessian type equations

Key Result

Theorem 2.1

Let the assumptions be given as above, and $\varphi$ is a $C^2$ solution to the equation eqn:main, then there exists a constant $C>0$ depending on the given parameters $n, A, K, p, \gamma$, and $\delta, R, \kappa_1, \kappa_2$ such that

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 2.1
  • Lemma 2.1: GPSS22GPS
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Definition 4.1
  • Theorem 4.1
  • Lemma 4.1