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

TL;DR

This paper proves Liouville theorems for fully nonlinear elliptic equations with small oscillation and periodic coefficients, showing solutions of quadratic growth are sums of quadratic polynomials and periodic functions.

Contribution

It extends Liouville theorems to fully nonlinear elliptic equations with periodic coefficients and small oscillation, generalizing previous linear and nonlinear results.

Findings

Established existence of quadratic growth solutions as sums of quadratic polynomials and periodic functions.

Proved Liouville type theorems under small oscillation conditions.

Generalized classical results to fully nonlinear elliptic equations with periodic data.

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.