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

Abstract

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