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)Δ∞Nu)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u_t-|Du|^{\\gamma }\\big (\\Delta u+(p-2)\\Delta _\\infty ^Nu\\big )=0 \\end{aligned}$$\\end{document}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 exists as a function and belongs to L^2_text {loc} 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}.