Smooth local solutions to weingarten equations and $σ_k$-equations

In this paper, we study the existence of smooth local solutions to Weingarten equations and $σ_k$-equations. We will prove that, for $2 \leq k \leq n$, the Weingarten equations and the $σ_k$-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.

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