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Hölder regularity for the parabolic perturbed fractional 1-Laplace equations

Dingding Li, Chao Zhang

Abstract

This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional $1$-Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the decomposition of the nonlocal integral into local and nonlocal components to handle their contributions separately. We aim to show the local Hölder continuity of weak solutions within the parabolic domain. More precisely, the solutions are spatially $α$-Hölder continuous with $0<α<\min\left\lbrace1, \frac{s_p p}{p-1} \right\rbrace$ and $γ$-Hölder continuous in time, where the value of $γ$ is determined by the fractional differentiability indexes $s_1$, $s_p$ and the exponent $p$. For both the super-quadratic case ($p\ge 2$) and the sub-quadratic case ($1<p<2$), we establish the Sobolev regularity of solutions, which underpins the derivation of Hölder continuity. All estimates are quantitative and depend only on the structural parameters of the equation. To the best of our knowledge, this is the first attempt to develop a regularity theory for such nonlocal parabolic equations.

Hölder regularity for the parabolic perturbed fractional 1-Laplace equations

Abstract

This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional -Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the decomposition of the nonlocal integral into local and nonlocal components to handle their contributions separately. We aim to show the local Hölder continuity of weak solutions within the parabolic domain. More precisely, the solutions are spatially -Hölder continuous with and -Hölder continuous in time, where the value of is determined by the fractional differentiability indexes , and the exponent . For both the super-quadratic case () and the sub-quadratic case (), we establish the Sobolev regularity of solutions, which underpins the derivation of Hölder continuity. All estimates are quantitative and depend only on the structural parameters of the equation. To the best of our knowledge, this is the first attempt to develop a regularity theory for such nonlocal parabolic equations.

Paper Structure

This paper contains 8 sections, 26 theorems, 207 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of related results.
  3. Main results
  4. Approach overview
  5. Preliminaries
  6. Sobolev regularity for the super-quadratic case
  7. Sobolev regularity for the sub-quadratic case
  8. Hölder regularity of solutions

Key Result

Theorem 1.3

Let $p>1$, $s_1,s_p\in(0,1)$ and suppose that $u$ is a locally bounded weak solution to problem 1.1 in the sense of Definition def1. Then, $u$ is locally Hölder continuous in $\Omega_I$. Moreover, for any $B_R(x_0)\subset\subset\Omega$ with $R\in(0,1)$, $r\in(0,R)$, $(t_0-2S,t_0)\subset\subset I$ wi where $\Gamma:=$

Figures (1)

  • Figure 1: 3D surfaces of $\Gamma(s_1, s_p)$ for $p=\frac{3}{2}$ (left) and $p=3$ (right).

Theorems & Definitions (46)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4: Embedding $W^{1,q}\hookrightarrow W^{\gamma,q}$
  • Lemma 2.5: Embedding $W^{\gamma,q}\hookrightarrow C^{0,\gamma-\frac{N}{q}}$
  • Lemma 2.6
  • ...and 36 more