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A note on the Liouville theorem of fully nonlinear elliptic equations

Dongsheng Li, Lichun Liang

TL;DR

The paper addresses the Liouville-type asymptotics for fully nonlinear elliptic equations $F(D^2u)=0$ in exterior domains, aiming to characterize solutions with at most quadratic growth without assuming $F$ is $C^2$ or restricting the dimension. It introduces a novel method that uses Evans-Krylov theory, barrier arguments, and Kelvin transformations to derive a sharp asymptotic expansion $u(x)=\frac{1}{2}x^{T}Ax+b\cdot x+d\,\Gamma(x)+c+\frac{e\cdot x}{|x^{T}(DF(A))^{-1}x|^{n/2}}+O(|x|^{1-n-\alpha})$ with a dimension-dependent correction $\Gamma(x)$. The result identifies the asymptotic quadratic form $A$, linear term $b$, and a Kelvin-type correction with coefficient $e$, together with logarithmic or power-like decay terms, and it holds under $F\in C^{1,2/n}$, convex, and uniformly elliptic, for viscosity solutions with quadratic growth. This advances the understanding of exterior-domain behavior for degenerate and fully nonlinear elliptic equations and extends prior works by removing reliance on higher regularity of $F$ and specific dimensional constraints.

Abstract

In this paper, a new method is presented to investigate the asymptotic behavior of solutions to the fully nonlinear uniformly elliptic equation $F(D^2u)=0$ in exterior domains. This method does not depend on the $C^2$ regularity of $F$ and the dimension $n$.

A note on the Liouville theorem of fully nonlinear elliptic equations

TL;DR

The paper addresses the Liouville-type asymptotics for fully nonlinear elliptic equations in exterior domains, aiming to characterize solutions with at most quadratic growth without assuming is or restricting the dimension. It introduces a novel method that uses Evans-Krylov theory, barrier arguments, and Kelvin transformations to derive a sharp asymptotic expansion with a dimension-dependent correction . The result identifies the asymptotic quadratic form , linear term , and a Kelvin-type correction with coefficient , together with logarithmic or power-like decay terms, and it holds under , convex, and uniformly elliptic, for viscosity solutions with quadratic growth. This advances the understanding of exterior-domain behavior for degenerate and fully nonlinear elliptic equations and extends prior works by removing reliance on higher regularity of and specific dimensional constraints.

Abstract

In this paper, a new method is presented to investigate the asymptotic behavior of solutions to the fully nonlinear uniformly elliptic equation in exterior domains. This method does not depend on the regularity of and the dimension .

Paper Structure

This paper contains 2 sections, 2 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['th1']}

Key Result

Theorem 1.1

Let $F\in C^{1,\frac{2}{n}}(\mathcal{S}^{n\times n})$ be uniformly elliptic and convex. Assume that $u$ is a viscosity solution of (eq1) and satisfies for some constant $C>0$. Then there exist some $A\in \mathcal{S}^{n\times n}$, $b, e \in \mathbb{R}^n$ and $c, d\in \mathbb{R}$ such that for any $0<\alpha<1$, where

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['th1']}
  • Remark 2.2