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Fractional $p$-caloric functions are Lipschitz

David Jesus, Aelson Sobral, José Miguel Urbano

Abstract

We study the parabolic fractional $p-$Laplace equation $$\p_t u+(-Δ_p)^su = 0$$ in the degenerate range \(2 \leq p < 2/(1-s)\). We show that weak solutions are Lipschitz continuous in space and, if \(p > 1/(1-s)\), also in time. We also prove a comparison principle for both weak and viscosity solutions, and establish the equivalence between the two notions of solution.

Fractional $p$-caloric functions are Lipschitz

Abstract

We study the parabolic fractional Laplace equation in the degenerate range \(2 \leq p < 2/(1-s)\). We show that weak solutions are Lipschitz continuous in space and, if \(p > 1/(1-s)\), also in time. We also prove a comparison principle for both weak and viscosity solutions, and establish the equivalence between the two notions of solution.
Paper Structure (15 sections, 23 theorems, 287 equations)

This paper contains 15 sections, 23 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Mathematical setup and preliminaries
  3. Notation
  4. Weak/viscosity solutions
  5. Known results and useful estimates
  6. Properties of weak and viscosity solutions
  7. Comparison principles
  8. Weak implies viscosity
  9. Viscosity implies weak
  10. Parabolic Lipschitz regularity
  11. The setup
  12. The cone of concavity
  13. Decomposition of the domain
  14. Proof of the Lipschitz regularity in space
  15. Time regularity and proof of Theorem \ref{['t:Lipschitz']}

Key Result

Theorem 1.1

Let $p \geq 2$ and $s \in (0,1)$ be such that Let $u$ be a weak solution to eq:parabolic-fractional-p-laplacian in $Q_1$, and assume that Set Then, the following assertions hold:

Theorems & Definitions (44)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • ...and 34 more