A systematic approach on the second order regularity of solutions to the general parabolic p -Laplace equation

We study a general form of a degenerate or singular parabolic equation u_t-|Du|^γ (Δ u+(p-2)Δ _∞ ^Nu )=0 that generalizes both the standard parabolic p -Laplace equation and the normalized version that arises from stochastic game theory. We develop a systematic approach to study second order Sobolev regularity and show that D^2u exists as a function and belongs to L^2_loc for a certain range of parameters. In this approach proving the estimate boils down to verifying that a certain coefficient matrix is positive definite. As a corollary we obtain, under suitable assumptions, that a viscosity solution has a Sobolev time derivative belonging to L^2_loc .