Regularity of viscosity solutions to fully nonlinear elliptic transmission problems

We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and regularity of solutions to flat interface transmission problems; and the $C^{0,\alpha }$, $C^{1,\alpha }$ and $C^{2,\alpha }$ regularity of viscosity solutions up to the transmission surface for the case of curved interfaces.