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On the modulus of continuity of solutions to nonlocal parabolic equations

Naian Liao

Abstract

A general modulus of continuity is quantified for locally bounded, local, weak solutions to nonlocal parabolic equations, under a minimal tail condition. Hölder modulus of continuity is then deduced under a slightly stronger tail condition. These regularity estimates are demonstrated under the framework of nonlocal $p$-Laplacian with measurable kernels.

On the modulus of continuity of solutions to nonlocal parabolic equations

Abstract

A general modulus of continuity is quantified for locally bounded, local, weak solutions to nonlocal parabolic equations, under a minimal tail condition. Hölder modulus of continuity is then deduced under a slightly stronger tail condition. These regularity estimates are demonstrated under the framework of nonlocal -Laplacian with measurable kernels.
Paper Structure (16 sections, 8 theorems, 175 equations)

This paper contains 16 sections, 8 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Novelty and Significance
  3. Definitions and Notation
  4. Energy Estimates
  5. Preliminary Tools
  6. Proof of Theorem \ref{['Thm:1:1']}: $1<p\le 2$
  7. Expansion of Positivity
  8. The First Step
  9. The Induction
  10. Modulus of Continuity
  11. Proof of Theorem \ref{['Thm:1:1']}: $p> 2$
  12. The First Alternative
  13. The Second Alternative
  14. The Induction
  15. Modulus of Continuity
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $u$ be a locally bounded, local, weak solution to Eq:1:1 in $E_T$ satisfying Eq:1:2 -- Eq:K with $p>1$. Then $u$ is locally continuous in $E_T$. More precisely, there exist constants $\boldsymbol\gamma>1$ and $\beta,\,\sigma\in(0,1)$ depending on the data $\{s, p, N, C_o, C_1\}$, such that for a provided the cylinders $(x_o,t_o)+Q_{R}(\boldsymbol\omega^{2-p})\subset (x_o,t_o)+Q_{\widetilde{R}}

Theorems & Definitions (15)

  • Theorem 1.1: general modulus of continuity
  • Theorem 1.2: Hölder modulus of continuity
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 5 more