$C^\infty$ local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$

TL;DR

This paper investigates the existence and convexity properties of smooth local solutions to the 2-Hessian equation in three-dimensional space, accommodating various behaviors of the right-hand side function, including sign changes and vanishing.

Contribution

It establishes the existence of smooth local solutions to the 2-Hessian equation in D, considering diverse conditions on the right-hand side function and analyzing their convexity.

Findings

Existence of $C^{}$ local solutions under various conditions on $f$.

Convexity properties of solutions related to the sign and zeros of $f$.

Associated linearized operators are uniformly elliptic.

Abstract

In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.