Homogeneous Solutions of Fully Nonlinear Elliptic Equations in Four Dimensions

TL;DR

This paper proves that in four dimensions, the only homogeneous solutions of order 2 to fully nonlinear uniformly elliptic equations are trivial, establishing a significant restriction on such solutions.

Contribution

It demonstrates the nonexistence of nontrivial homogeneous order 2 solutions in four-dimensional fully nonlinear elliptic equations, advancing understanding of solution structures.

Findings

No nontrivial homogeneous order 2 solutions in 4D

Restricts solution forms for fully nonlinear elliptic equations

Enhances classification of elliptic PDE solutions

Abstract

We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.