Solvability and Galerkin Approximations of a Class of Nonlinear Operator Equations

Abstract

We generalize the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints. The corresponding operator equation has a dual-dual type structure since the nonlinear operator involved has itself a dual structure with a strongly monotone and Lipschitz-continuous main operator. We provide sufficient conditions for the existence and uniqueness of solution of the continuous and Galerkin formulations, and derive a Strang-type estimate for the associated error. An application to the coupling of mixed-FEM and BEM for a nonlinear transmission problem in potential theory is also described.