Regularity of elliptic transmission problems and a new family of integro-differential operators related to the Monge-Ampère equation

This dissertation is divided into two main topics. First, we study transmission problems for elliptic equations, both linear and nonlinear, and prove existence, uniqueness, and optimal regularity of solutions. In our first work, we consider a problem for harmonic functions and use geometric techniques. Our second work considers viscosity solutions to fully nonlinear transmission problems. Given the nonlinear nature of these equations, our arguments are based on perturbation methods and comparison principles. The second topic is related to nonlocal Monge-Ampère equations. We define a new family of integro-differential equations arising from geometric considerations and study some of their properties. Furthermore, we consider a Poisson problem in the full space and prove existence, uniqueness, and C¹,¹ regularity of solutions. For this problem, we use tools from convex analysis and symmetrization.