Smooth local solutions to weingarten equations and $\sigma_k$-equations

TL;DR

This paper proves the existence of smooth local solutions to Weingarten and $\sigma_k$-equations for all relevant k, regardless of the sign of the right-hand side, by establishing uniform ellipticity of linearized equations.

Contribution

It establishes the existence of smooth local solutions to Weingarten and $\sigma_k$-equations for all k between 2 and n, regardless of the sign of the right-hand side.

Findings

Smooth local solutions exist for Weingarten and $\sigma_k$-equations.

Linearized equations are uniformly elliptic with appropriate initial solutions.

Existence holds for all $2 \\leq k \\leq n$ regardless of sign.

Abstract

In this paper, we study the existence of smooth local solutions to Weingarten equations and $\sigma_k$-equations. We will prove that, for $2 \leq k \leq n$, the Weingarten equations and the $\sigma_k$-equations always have smooth local solutions regardless of the sign of the functions in the right-hand side of the equations. We will demonstrate that the associate linearized equations are uniformly elliptic if we choose the initial approximate solutions appropriately.