C1,1 regularity for degenerate complex Monge-Ampere equations and geodesic rays

We prove a C-1,C-1 estimate for solutions of complex Monge-Ampere equations on compact Kahler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local C-1,C-1 regularity of geodesic rays in the space of Kahler metrics associated to a test configuration, as well as the local C-1,C-1 regularity of quasi-psh envelopes in nef and big classes away from the non-Kahler locus.