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 $$\begin{aligned} u_t-|Du|^{\gamma }\big (\Delta u+(p-2)\Delta _\infty ^Nu\big )=0 \end{aligned}$$ u t - | D u | γ ( Δ u + ( p - 2 ) Δ ∞ 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 $$D^2u$$ D 2 u exists as a function and belongs to $$L^2_\text {loc}$$ L loc 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 $$L^2_\text {loc}$$ L loc 2 .