Our study delves into the existence and regularity of  transmission problems represented by the equation \( \sigma_i(Du)  F(D^2u)=f \), concerning regions where \(u>0\) and \(u<0\). This work solves  the question on existence of solutions for the problem by utilizing a comparison principle for an auxiliary problem and Perron's Method.  Furthermore, we establish \(C^1\) regularity estimates for the interior as a subsequent step in our analysis. 

This is joint work with Edgard Pimentel.