Well-posedness for a transmission problem connecting first and second-order operators

We establish the existence and uniqueness of viscosity solutions within a domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ for a class of equations governed by elliptic and eikonal type equations in disjoint regions. Our primary motivation stems from the Hamilton-Jacobi equation that arises in the context of a stochastic optimal control problem.