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In this paper, we study the existence of smooth local solutions to Weingartenequations 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 localsolutions regardless of the sign of the functions in the right-hand side of theequations. We will demonstrate that the associate linearized equations areuniformly elliptic if we choose the initial approximate solutionsappropriately.

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