Symmetry of solutions of minimal gradient graph equations on punctured space

TL;DR

This paper investigates the symmetry and existence of solutions for minimal gradient graph equations on punctured space, extending classification results and characterizing solvability based on asymptotic behavior.

Contribution

It extends classification results for Monge-Ampère equations to a broader class of minimal gradient graph equations on punctured space.

Findings

Symmetry results for solutions on punctured space

Existence conditions for solutions based on asymptotic behavior

Extension of classification results to new equations

Abstract

In this paper, we study symmetry and existence of solutions of minimal gradient graph equations on punctured space $\mathbb R^n\setminus\{0\}$, which include the Monge-Amp\`ere equation, inverse harmonic Hessian equation and the special Lagrangian equation. This extends the classification results of Monge-Amp\`ere equations. Under some conditions, we also give the characterization of the solvability on exterior Dirichlet problem in terms of their asymptotic behaviors.