  We study PDE of the form $\max\{F(D^2u,x)-f(x), H(Du)\}=0$ where $F$ is
uniformly elliptic and convex in its first argument, $H$ is convex, $f$ is a
given function and $u$ is the unknown. These equations are derived from dynamic
programming in a wide class of stochastic singular control problems. In
particular, examples of these equations arise in mathematical finance models
involving transaction costs, in queuing theory, and spacecraft control
problems. The main aspects of this work are to identify conditions under which
solutions are uniquely defined and have Lipschitz continuous gradients. We also
generalize previous results known for the case where $M\mapsto F(M,x)$ is the
maximum of finitely many linear functions.
