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

2023/03/31 by Feng, Yawen, Parviainen, Mikko, Sarsa, Saara · 1 citation
#35B65 #35K55 #35K65 #35K67 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.00108

Abstract

We study a general form of a degenerate or singular parabolic equation u_t-|Du|^γ(Δu+(p-2)Δ_∞^Nu)=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 L²_{\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 L²_{\text loc}.

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