Second order Sobolev regularity results for the generalized p-parabolic equation

We study a general class of parabolic equations u t − | D u | γ ( Δ u + ( p − 2 ) Δ ∞ N u ) = 0 , which can be highly degenerate or singular. This class contains as special cases the standard parabolic p -Laplace equation and the normalized version that arises from stochastic game theory. Utilizing the systematic approach developed in our previous work we establish second order Sobolev regularity together with a priori estimates and improved range of parameters. In addition we derive second order Sobolev estimate for a nonlinear quantity. This quantity contains many useful special cases. As a corollary we also obtain that a viscosity solution has locally L 2 -integrable Sobolev time derivative.