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

In this paper, we study the existence of smooth local solutionsto Weingarten equations and $\sigma_k$-equations. We will prove that, for $2\le k\le n-1$,the Weingarten equations and the $\sigma_k$-equationsalways have smooth local solutions regardless ofthe sign of the functions in the right-hand side of the equations.We will demonstrate that the associated linearized equations are uniformly ellipticif we choose the initial approximate solutions appropriately.