A PRIORI ESTIMATES FOR INFINITELY DEGENERATE QUASILINEAR EQUATIONS

We prove a priori bounds for derivatives of solutions w of a class of quasilinear equations of the form div A(x,w)Delta w + gamma(->)(x, w) . V w f (x,w) = 0, where x = (x(1),..., x(n)), and where f, gamma(->) = (gamma(i))1 <= i <= n, and A = (a(ij))1 <= i,j <= are C-infinity. The rank of the square symmetric matrix A is allowed to degenerate, as all but one eigenvalue of A are permitted to vanish to infinite order. We estimate derivatives of w of arbitrarily high order in terms of just w and its first derivatives. These estimates will be applied in a subsequent work to establish existence, uniqueness and regularity of weak solutions of the Dirchlet problem.