Regularity for quasilinear operators with infinite vanishing ellipticity

We prove a priory regularity for solutions of a class of quasilinear equations which fail to be elliptic or even subelliptic since eigenvalues are allowed to vanish to any order. We obtain existence and uniqueness of solutions for the Dirichlet problem and regularity of weak continuous solutions as consequences. These results are new even in the linear case. In two dimensions the quasilinear theorems can be applied to obtain regularity of solutions of infinite degenerate Monge-Ampere equations. This is a project conducted in conjunction with Eric Sawyer from McMaster University, and Richard Wheeden from Rutgers University.