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Liouville theorem for fully nonlinear elliptic equations with the small oscillation and the periodicity in $x$ and the periodic right hand term

Lichun Liang

Abstract

In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u,x)=f$ in $\mathbb{R}^n$, where $f$ is periodic and $F$ has the periodicity in $x$. Under the assumption that the oscillation of $F(M,x)$ in $x$ is ``small", we establish the existence and Liouville type results for quadratic growth solutions, which can be expressed into the sum of a quadratic polynomial and a periodic function. Consequently, these results are generalization of the existing results for linear elliptic equations $a_{ij}D_{ij}u=0$ and fully nonlinear elliptic equations $F(D^2u)=f$ with the periodic data.

Liouville theorem for fully nonlinear elliptic equations with the small oscillation and the periodicity in $x$ and the periodic right hand term

Abstract

In this paper, we study quadratic growth solutions of fully nonlinear elliptic equations of the form in , where is periodic and has the periodicity in . Under the assumption that the oscillation of in is ``small", we establish the existence and Liouville type results for quadratic growth solutions, which can be expressed into the sum of a quadratic polynomial and a periodic function. Consequently, these results are generalization of the existing results for linear elliptic equations and fully nonlinear elliptic equations with the periodic data.
Paper Structure (5 sections, 13 theorems, 143 equations)

This paper contains 5 sections, 13 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Uniformly elliptic equations in the whole space
  3. Existence
  4. Liouville type result
  5. Uniformly elliptic equations on exterior domains

Key Result

Theorem 1.2

Let $F\in C^2(\mathcal{S}^{n\times n}\times \mathbb{R}^n)$ satisfy (H1), (H2) and (H3) and $f\in C^{\alpha}(\mathbb{R}^n)$ be periodic for some $\alpha\in (0,1)$. Assume that there exists a constant $\bar{C}>0$ such that for all $r\leq 2$ and $x_0\in Q_1$. Then for any $A\in \mathcal{S}^{n\times n}$, has a unique solution $v\in C^{2}(\mathbb{R}^n)\cap \mathbb{T}$.

Theorems & Definitions (31)

  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Definition 1
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • ...and 21 more