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Lipschitz regularity for parabolic fractional $p$-Laplace equations

Harsh Prasad

Abstract

We prove that local weak solutions to nonlocal parabolic $p$-Laplace equations are locally Lipschitz continuous in space, uniformly in time for every $1<p<\infty$ and $s \in (0,1)$ whenever $sp > p-1$. Our results hold for symmetric, translation-invariant kernels satisfying standard ellipticity bounds, including kernels that may be discontinuous and require only that the tail of the solution be bounded. In the linear case, our proof provides a different route avoiding blow up arguments and Liouville theorems.

Lipschitz regularity for parabolic fractional $p$-Laplace equations

Abstract

We prove that local weak solutions to nonlocal parabolic -Laplace equations are locally Lipschitz continuous in space, uniformly in time for every and whenever . Our results hold for symmetric, translation-invariant kernels satisfying standard ellipticity bounds, including kernels that may be discontinuous and require only that the tail of the solution be bounded. In the linear case, our proof provides a different route avoiding blow up arguments and Liouville theorems.
Paper Structure (17 sections, 20 theorems, 74 equations, 1 figure)

This paper contains 17 sections, 20 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and Novelty
  3. Main Theorem
  4. Extensions
  5. Outline
  6. Preliminaries
  7. Fractional Sobolev spaces.
  8. Tail space.
  9. Hölder seminorms.
  10. The Touching Lemma
  11. Scaling and Normalisation
  12. Proof of Main Theorem
  13. The Ishii--Lions framework
  14. The master inequality
  15. Integral estimates
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $u$ be a local weak solution to eq:main-eq. Suppose $(x_0, t_0) \in \Omega \times I$ and $R > 0$ are such that $Q_{2R,2R^{sp}}(x_0,t_0) \Subset \Omega \times I$. Then $u(\cdot, t)$ is locally Lipschitz in space, uniformly in time and with $M_R$ as in eq:MR-def, the following estimates hold for a The constant $C > 0$ depends only on $N, s, p, \lambda, \Lambda$.

Figures (1)

  • Figure 1: Geometric decomposition of the integration domain $B_1$ into regions $C, D_1, D_2,$ and $D_3$, oriented along the unit vector $\hat{a}_\nu = a_\nu / |a_\nu|$.

Theorems & Definitions (47)

  • Theorem 1.1: Spatial Lipschitz regularity
  • Remark 1: Sharpness of the condition
  • Remark 2: Time regularity
  • Remark 3: Discontinuous kernels
  • Definition 1
  • Proposition 1: Local boundedness
  • proof
  • Proposition 2: Hölder continuity
  • proof
  • Lemma 1
  • ...and 37 more