In this talk we extend Escobar's classification to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere with prescribed constant mean curvature on its boundary, and annular domains with minimal boundary. If time permits, we also present the classification in the 2-dimensional case.  This is joint work with Jose Espinar (IMPA).