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Fine boundary regularity for the singular fractional p-Laplacian

Antonio Iannizzotto, Sunra Mosconi

Abstract

We study the boundary weighted regularity of weak solutions $u$ to a $s$-fractional $p$-Laplacian equation in a bounded smooth domain $Ω$ with bounded reaction and nonlocal Dirichlet type boundary condition, in the singular case $p\in(1,2)$ and with $s\in(0,1)$. We prove that $u/{\rm d}_Ω^s$ has a $α$-Hölder continuous extension to the closure of $Ω$, ${\rm d}_Ω(x)$ meaning the distance of $x$ from the complement of $Ω$. This result corresponds to that of ref. [28] for the degenerate case $p\ge 2$.

Fine boundary regularity for the singular fractional p-Laplacian

Abstract

We study the boundary weighted regularity of weak solutions to a -fractional -Laplacian equation in a bounded smooth domain with bounded reaction and nonlocal Dirichlet type boundary condition, in the singular case and with . We prove that has a -Hölder continuous extension to the closure of , meaning the distance of from the complement of . This result corresponds to that of ref. [28] for the degenerate case .
Paper Structure (16 sections, 19 theorems, 277 equations, 2 figures)

This paper contains 16 sections, 19 theorems, 277 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Related results
  4. Motivations and applications
  5. Sketch of the proof
  6. Plan of the paper
  7. Preliminaries
  8. Properties of the distance function
  9. Functional setting
  10. Optimal regularity up to the boundary
  11. Torsion functions and barriers
  12. Lower bound
  13. Upper bound
  14. Oscillation estimate and conclusion
  15. Some elementary inequalities
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $p>1$, $s\in(0,1)$, $\Omega\subset{\mathbb R}^N$ be a bounded domain with a $C^{1,1}$-smooth boundary $\partial\Omega$. Then, there exist $\alpha\in(0,s)$, $C>0$, depending on $N$, $p$, $s$, and $\Omega$, with the following property: for all $f\in L^\infty(\Omega)$, if $u\in W^{s,p}_0(\Omega)$ i

Figures (2)

  • Figure 1: The ball $\tilde{B}_{x, R}$, with center on the normal direction.
  • Figure 2: The regularized set $A_{R}$ in gray satisfies $D_{3R/4}\subset A_R\subset D_R$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Theorem 2.7
  • ...and 22 more