In this paper, the Dirichlet problem is studied for degenerate nonlinear Bellman equations. The main result is an estimate on the second mixed derivative of the solution on the boundary. In some cases this estimate yields estimates on all second derivatives both inside and on the boundary. As an example, the elementary Monge-Ampère equation is studied on a smooth strictly convex domain, and the existence of a solution smooth up to the boundary is established. The method of estimating the second mixed derivatives is based on the reduction to an estimate of the first derivatives of the solution of an auxiliary equation on a suitable closed manifold without boundary.