Singular Solutions of Hessian Elliptic Equations in Five Dimensions

TL;DR

This paper constructs specific homogeneous solutions to Hessian elliptic equations in five dimensions, demonstrating the existence of solutions with particular regularity properties outside the origin.

Contribution

It establishes the existence of homogeneous solutions of order 2−δ for any δ in [0,1) to uniformly elliptic Hessian equations in five dimensions, expanding understanding of solution regularity.

Findings

Existence of solutions with order 2−δ for δ in [0,1)

Solutions are analytic outside zero

Results apply to uniformly elliptic Hessian equations in R^5

Abstract

We show that for any $\delta\in [0,1)$ there exists a homogeneous order $2-\delta$ analytic outside zero solution to a uniformly elliptic Hessian equation in R^5.