A class of nonlinear elliptic and parabolic optimal control problems with mixed control- state constraints is considered. Extending a method known for the control of ordinary differential equations to the case of PDEs, the Yosida - Hewitt theorem is applied to show that the Lagrange multipliers are functions of certain Lp- spaces. By bootstrapping arguments, under natural assumptions, optimal controls are shown to be Lipschitz continuous in the elliptic case and Holder continuous for parabolic problems.