On the Regularity Theory of Fully Nonlinear Parabolic Equations

Recently M. Crandall and P. L. Lions [3] developed a very successful method for proving the existence of solutions of nonlinear second-order partial differential equations. Their method, called the theory of viscosity solutions, also applies to fully nonlinear equations (in which even the second order derivatives can enter in nonlinear fashion). Solutions produced by the viscosity method are guaranteed to be continuous, but not necessarily smooth. Here we announce smoothness results for viscosity solutions. Our methods extend those of [1]. We obtain Krylov-Safonov (i.e. C estimates [8]), C 1 ' " , Schauder (C) and W estimates for viscosity solutions of uniformly parabolic equations in general form. The results can be viewed as a priori estimates on the classical C solutions. Our method produces, in particular, regularity results for a broad new array of nonlinear heat equations, including the Bellman equation [6]: