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Fine Boundary Regularity For The Fractional (p,q)-Laplacian

R. Dhanya, Ritabrata Jana, Uttam Kumar, Sweta Tiwari

TL;DR

This work advances the boundary regularity theory for the fractional nonlocal operator $(-\Delta)_p^s+(-\Delta)_q^s$ by proving that weak solutions satisfy $\frac{u}{d_{\Omega}^{s}}\in C^{\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$ when $f$ is nonnegative and bounded. The authors develop a nonlocal boundary Harnack approach complemented by a novel barrier construction that does not rely on the operator’s homogeneity, enabling fine boundary regularity results for the nonhomogeneous fractional $(p,q)$-Laplacian. They introduce a nonlocal excess quantity $\mathrm{Ex}(u,m,R)$ to quantify boundary oscillations and derive sharp lower/upper bounds via barrier arguments and obstacle problems. The framework is extended to sign-changing data for a restricted range $s\in(0,1/q)$, establishing analogous Hölder regularity and highlighting the method’s robustness beyond nonhomogeneous, homogeneous, or scaling-invariant settings. These results contribute to a deeper understanding of boundary behavior in nonlocal double-phase problems and have implications for related Pohozaev identities and variational analyses.

Abstract

In this article, we deal with the fine boundary regularity, a weighted Hölder regularity of weak solutions to the problem involving the fractional $(p,q)$ Laplacian denoted by $(-Δ)_{p}^{s} u + (-Δ)_{q}^{s} u = f(x)$ in $Ω,$ and $u=0$ in $\mathbb{R}^N\setminusΩ;$ where $Ω$ is a $C^{1,1}$ bounded domain and $2 \leq p \leq q <\infty.$ For $0<s<1$ and for non-negative data $f\in L^{\infty}(Ω),$ we employ the nonlocal analogue of the boundary Harnack method to establish that $u/{d_Ω^{s}} \in C^α(\BarΩ)$ for some $α\in (0,1),$ where $d_Ω(x)$ is the distance of $x$ from the boundary. A novel barrier construction allows us to analyse the regularity theory even in the absence of the scaling or the homogeneity properties of the operator. Additionally, we extend our idea to sign changing bounded $f$ as well and prove a fine boundary regularity for fractional $(p,q)$ Laplacian for some range of $s.$

Fine Boundary Regularity For The Fractional (p,q)-Laplacian

TL;DR

This work advances the boundary regularity theory for the fractional nonlocal operator by proving that weak solutions satisfy for some when is nonnegative and bounded. The authors develop a nonlocal boundary Harnack approach complemented by a novel barrier construction that does not rely on the operator’s homogeneity, enabling fine boundary regularity results for the nonhomogeneous fractional -Laplacian. They introduce a nonlocal excess quantity to quantify boundary oscillations and derive sharp lower/upper bounds via barrier arguments and obstacle problems. The framework is extended to sign-changing data for a restricted range , establishing analogous Hölder regularity and highlighting the method’s robustness beyond nonhomogeneous, homogeneous, or scaling-invariant settings. These results contribute to a deeper understanding of boundary behavior in nonlocal double-phase problems and have implications for related Pohozaev identities and variational analyses.

Abstract

In this article, we deal with the fine boundary regularity, a weighted Hölder regularity of weak solutions to the problem involving the fractional Laplacian denoted by in and in where is a bounded domain and For and for non-negative data we employ the nonlocal analogue of the boundary Harnack method to establish that for some where is the distance of from the boundary. A novel barrier construction allows us to analyse the regularity theory even in the absence of the scaling or the homogeneity properties of the operator. Additionally, we extend our idea to sign changing bounded as well and prove a fine boundary regularity for fractional Laplacian for some range of
Paper Structure (8 sections, 33 theorems, 274 equations)

This paper contains 8 sections, 33 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some Important Estimates
  4. The Lower Bound
  5. The Upper Bound
  6. Weighted Hölder Regularity For Bounded Non-negative Data
  7. Weighted Hölder Regularity For Bounded sign changing Data
  8. Introduction

Key Result

Theorem 1.1

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ with $C^{1,1}$ boundary and $\mathrm{d}_{\Omega}(x)$ is defined in disdef. Let $u\in W^{s,p}_0(\Omega)\cap W^{s,q}_0(\Omega)$ be the weak solution of where $2 \leqslant p \leqslant q<\infty.$ Assume further that any of the following two conditions are satisfied: Then, $\frac{u}{\mathop{\mathrm{d^{s}_{\Omega}}}\nolimits}\in C^\alpha(\bar{\Omega})

Theorems & Definitions (46)

  • Theorem 1.1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 36 more