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 $$u_t-\mathrm{\mid }Du{\mathrm{\mid }}^{\gamma }(\mathrm{\Delta }u+(p-2){\mathrm{\Delta }}_{\mathrm{\infty }}^{N}u)=0$$ 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 $D^{2}u$ exists as a function and belongs to $L_{\text{l}oc}^2$ 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 $L_{\text{l}oc}^2$.