Nonexistence of nontrivial generalized solutions for 2-D and 3-D BVPs with nonlinear mixed type equations

A brief survey of known results, open problems and new contributions to the understanding of the nonexistence of nontrivial solutions to nonlinear boundary value problems (BVPs) whose linear part is of mixed elliptic-hyperbolic type is given. Crucial issues discussed include: the role of so-called critical growth of the nonlinear terms in the equation (often related to threshold values of continuous and compact embedding for Sobolev spaces in Lebesgue spaces), the role that hyperbolicity in the principal part plays in over-determining solutions with classical regularity if data is prescribed everywhere on the boundary, the relative lack of regularity that solutions to such problems possess and the subsequent importance to address nonexistence of generalized solutions.