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Local Hölder regularity for nonlocal parabolic $p$-Laplace equations

Karthik Adimurthi, Harsh Prasad, Vivek Tewary

Abstract

We prove local Hölder regularity for a nonlocal parabolic equations of the form \begin{align*} \partial_t u + \text{P.V.}\int_{\mathbb{R}^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+sp}}\,dy=0, \end{align*} for $p\in (1,\infty)$ and $s \in (0,1)$.

Local Hölder regularity for nonlocal parabolic $p$-Laplace equations

Abstract

We prove local Hölder regularity for a nonlocal parabolic equations of the form \begin{align*} \partial_t u + \text{P.V.}\int_{\mathbb{R}^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+sp}}\,dy=0, \end{align*} for and .
Paper Structure (21 sections, 14 theorems, 100 equations)

This paper contains 21 sections, 14 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. A brief history of the problem
  3. On historical development of intrinsic scaling
  4. Notations and Auxiliary Results
  5. Notations
  6. Structure of the equation
  7. Function spaces
  8. Definition of weak solution
  9. Auxiliary Results
  10. Main results
  11. Preliminary Results
  12. Energy estimates
  13. Shrinking Lemma
  14. Tail Estimates
  15. De Giorgi iteration Lemma
  16. ...and 6 more sections

Key Result

Theorem 2.2

Let $t_2>t_1>0$ and suppose $s\in(0,1)$ and $1\leq p<\infty$. Then for any $f\in L^p(t_1,t_2;W^{s,p}(B_r))\cap L^\infty(t_1,t_2;L^2(B_r))$, we have

Theorems & Definitions (42)

  • Remark 1.1
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 3.1
  • proof
  • ...and 32 more