On the regularity of solutions of a degenerate parabolic Bellman equation (Q1081754)

We consider the parabolic Bellman equation of the type: \[ u_ t+\max \{Lu-f,du-g\}=0\quad in\quad Q_ T\quad u=0\quad on\quad \partial_ pQ_ T, \] where f, g and d are given functions and L is a second order uniformly elliptic operator. This equation can be regarded as the time- dependent obstacle problem. Our main result is that there exists a unique solution \[ u\in L^{\infty}_{loc}([0,T);W^{2,\infty}(\Omega))\cap W^{\infty}_{loc}([0,T);L^{\infty}(\Omega)) \] under some assumptions. In order to prove the theorem we use the elliptic regularization and penalization.