Surfaces of prescribed linear Weingarten curvature in R3

Given a, b is an element of R and Phi is an element of C-1 (S-2), we study immersed oriented surfaces Sigma in the Euclidean 3-space R-3 whose mean curvature H and Gauss curvature K satisfy 2aH + bK = Phi(N), where N : Sigma -> S-2 is the Gauss map. This theory widely generalizes some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function 4), we exhibit a classification result for rotational surfaces in the case that. the underlying fully nonlinear PDE that governs these surfaces is elliptic or hyperbolic.