  In this paper we prove Holder regularity of the gradient for solutions of
Dirichlet problem associate to degenerate elliptic equations, extending the
recent result of Imbert and Silvestre. Indeed we obtain regularity up to the
boundary and when the equation has lower order terms.The proof follows their
scheme but requires new tools and new ideas. In particular we give some a
priori Lipschitz and H\"older estimates in the presence of boundary condition
on one part of the boundary.
