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Local Boundedness of a Direction Dependent Double Phase Nonlocal Elliptic Equation

Hamid El Bahja

TL;DR

This work addresses local boundedness for weak solutions to a direction-dependent, anisotropic, nonlocal double-phase elliptic equation with exponents $(s_i,p)$ and $(t_i,q)$. It develops anisotropic fractional Sobolev spaces and tail notions, proving a Sobolev–Poincaré embedding and a localized Poincaré inequality to support energy estimates. A nonlocal Caccioppoli inequality is established and combined with a De Giorgi–type iteration to prove that weak solutions are locally bounded under natural relations among the exponents and a tail condition. The results advance regularity theory for anisotropic, nonlocal, double-phase problems, with local boundedness shown but local continuity left as an open problem; tail behavior and directional interactions are central to the analysis.

Abstract

This paper investigates the local boundedness of weak solutions to a direction-dependent double-phase nonlocal elliptic equation. By employing refined energy estimates and De Giorgi-type techniques, we establish the local boundedness of these solutions.

Local Boundedness of a Direction Dependent Double Phase Nonlocal Elliptic Equation

TL;DR

This work addresses local boundedness for weak solutions to a direction-dependent, anisotropic, nonlocal double-phase elliptic equation with exponents and . It develops anisotropic fractional Sobolev spaces and tail notions, proving a Sobolev–Poincaré embedding and a localized Poincaré inequality to support energy estimates. A nonlocal Caccioppoli inequality is established and combined with a De Giorgi–type iteration to prove that weak solutions are locally bounded under natural relations among the exponents and a tail condition. The results advance regularity theory for anisotropic, nonlocal, double-phase problems, with local boundedness shown but local continuity left as an open problem; tail behavior and directional interactions are central to the analysis.

Abstract

This paper investigates the local boundedness of weak solutions to a direction-dependent double-phase nonlocal elliptic equation. By employing refined energy estimates and De Giorgi-type techniques, we establish the local boundedness of these solutions.

Paper Structure

This paper contains 4 sections, 7 theorems, 81 equations.

Table of Contents

  1. Introduction and statement of main results
  2. Auxiliary results
  3. Existence of weak solution
  4. Caccioppoli estimates and local boundedness

Key Result

Theorem 1.1

Let the assumptions (1.3), (1.5), and (1.9) hold, and assume further that Then, every weak solution $u \in A(\mathbb{R}^{N}, \mathbb{R}) \cap L^{q-1}_{\vec{s}, p}(\mathbb{R})$ to (1.1) is locally bounded.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Theorem 3.2
  • ...and 4 more