Hypoellipticity for infinitely degenerate quasilinear equations and the dirichlet problem

In [10], we considered a class of infinitely degenerate quasilinear equations of the form div $$A(x,w)\nabla w + \overrightarrow r (x,w) \cdot \nabla w + f(x,w) = 0$$ and derived a priori bounds for high order derivatives Daw of their solutions in terms of w and ▿w. We now show that it is possible to obtain bounds in terms of just w 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 $${\varrho ^\infty }$$-regularity of solutions for the corresponding Dirichlet problem with continuous boundary data.