Hypoellipticity for infinitely degenerate quasilinear equations and the   Dirichlet problem
  In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds just in terms of the supremum norm for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness and interior regularity of solutions for the Dirichlet problem with continuous boundary data. 
Cristian Rios, Eric Sawyer and Richard Wheeden
math.AP