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

Abstract

We study a general class of parabolic equations$u_t-{\left|Du\right|}^{\gamma }(\Delta u+(p-2\left){\Delta }_{\infty }^{N}u\right)=0,$ 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.