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

Abstract

We study a general form of a degenerate or singular parabolic equation ut−|Du|γ(Δu+(p−2)ΔN∞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 D2u exists as a function and belongs to L2loc 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 L2loc.