Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Second order Sobolev regularity for normalized parabolic $p(x)$-Laplace equations via the algebraic structure

Yuqing Wang, Yizhe Zhu

Abstract

Denote by $Δ$ the Laplacian and by $Δ_\infty$ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $ΔvΔ_\infty v$: for every $v\in C^{\infty}$, $$\bigg| |D^2vDv|^2-ΔvΔ_\infty v-\frac{1}{2}[|D^2v|^2-(Δv)^2]|Dv|^2\bigg| \le\frac{n-2}{2}[|D^2v|^2|Dv|^2-|D^2vDv|^2]$$ Based on this, we prove the result: When $n\ge2$ and $p(x)\in(1,2)\cup(2,3+\frac{2}{n-2})$, the viscosity solutions to parabolic normalized $p(x)$-Laplace equation have the $W^{2,2}_{loc}$-regularity in the spatial variable and the $W^{1,2}_{loc}$-regularity in the time variable.

Second order Sobolev regularity for normalized parabolic $p(x)$-Laplace equations via the algebraic structure

Abstract

Denote by the Laplacian and by the -Laplacian. A fundamental inequality is proved for the algebraic structure of : for every , Based on this, we prove the result: When and , the viscosity solutions to parabolic normalized -Laplace equation have the -regularity in the spatial variable and the -regularity in the time variable.
Paper Structure (4 sections, 14 theorems, 82 equations)

This paper contains 4 sections, 14 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The estimate of normalized parabolic $p(x)$-Laplacian
  4. The estimate of normalized parabolic $p(x)$-Laplacian

Key Result

Theorem 1.1

Let $n\ge2$, $f\in W^{1,\infty}(\Omega)$, $p(x,t)\in C^1(\Omega_T)$ and $p(x,t)\in(1,3+\frac{2}{n-2})$. Then for any vicosity solution $u=u(x,t)$ to we have $u_t$, $D^2u\in L^2_{loc}(\Omega_T)$. Moreover, $D(|Du|^{\frac{p(x)-2}{2}}Du)\in L_{loc}^2(\Omega_T)$ and

Theorems & Definitions (25)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • proof : Proof of Theorem 1.1.
  • Lemma 3.3
  • ...and 15 more