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Boundary behavior of solutions to fractional $p$-Laplacian equation

Alireza Ataei

TL;DR

The paper investigates boundary behavior for solutions to the nonlocal nonlinear equation $(-\Delta_p)^s u=f$ on bounded domains. It develops a generalized Hopf's lemma and a global boundary Harnack principle using Wiener regularity, the torsion function $u_{\text{tor}}$, and viscosity-weak formulations, and extends Wiener criteria to nonzero right-hand sides via perturbation. A central contribution is establishing the isolation of the first $(s,p)$-eigenvalue $\Lambda_{p,q}$ under Wiener regularity (with a noted exception when $q=p$). These results deepen the understanding of boundary regularity and spectral properties for fractional $p$-Laplacians and have implications for the qualitative behavior of nonlocal nonlinear elliptic equations.

Abstract

In this work, a generalized Hopf's lemma and a global boundary Harnack inequality are proved for solutions to fractional $p$-Laplacian equations. Then, the isolation of the first $(s,p)$-eigenvalue is shown in bounded open sets satisfying the Wiener criterion.

Boundary behavior of solutions to fractional $p$-Laplacian equation

TL;DR

The paper investigates boundary behavior for solutions to the nonlocal nonlinear equation on bounded domains. It develops a generalized Hopf's lemma and a global boundary Harnack principle using Wiener regularity, the torsion function , and viscosity-weak formulations, and extends Wiener criteria to nonzero right-hand sides via perturbation. A central contribution is establishing the isolation of the first -eigenvalue under Wiener regularity (with a noted exception when ). These results deepen the understanding of boundary regularity and spectral properties for fractional -Laplacians and have implications for the qualitative behavior of nonlocal nonlinear elliptic equations.

Abstract

In this work, a generalized Hopf's lemma and a global boundary Harnack inequality are proved for solutions to fractional -Laplacian equations. Then, the isolation of the first -eigenvalue is shown in bounded open sets satisfying the Wiener criterion.
Paper Structure (9 sections, 21 theorems, 139 equations)

This paper contains 9 sections, 21 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spaces
  4. Fractional Poincaré-Sobolev inequality and fractional Sobolev embedding
  5. Notions of solutions
  6. Wiener criterion
  7. Hopf's lemma and Global boundary Harnack inequality
  8. Eigenvalue problem
  9. acknowledgement

Key Result

Lemma 1.1

Let $u \in \operatorname{L}^{p-1}_{ps}(\mathbb{R}^n) \cap C(\overline \Omega_{\delta})$ be a non-negative function for a $\delta >0$ and $K \Subset \Omega$. Assume that $(-\Delta_p)^s u \geq f$ in $\Omega$ in the viscosity sense, where $f \in C(\Omega)$ satisfies Then, $u>0$ in $\Omega$ and for a constant $C>0.$

Theorems & Definitions (46)

  • Lemma 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • ...and 36 more