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

2024/04/09 by Yawen Feng, Feng, Yawen, Mikko Parviainen +3 · 2 citations
Computer Science · Mathematics · #35B65 #35K55 #35K65 #35K67 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2404.06161

openalex publication_date 2024/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a general class of parabolic equations u_t-|Du|^γ(Δu+(p-2) Δ_∞^N u)=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 L²-integrable Sobolev time derivative.

Cited by

• Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations
• Boundary Hölder gradient estimates for parabolic p-Laplace type equations

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