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 $$u_t-|Du|^{\gamma}\big(\Delta u+(p-2)\Delta_\infty^Nu\big)=0$$ 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 $L^2_{\text loc}$ 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 $L^2_{\text loc}$.