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On sliding methods for mixed local and nonlocal equations and Gibbons' conjecture

Yinbin Deng, Pengyan Wang, Zhihao Wang, Leyun Wu

TL;DR

The paper develops a refined sliding framework to study equations with mixed local and nonlocal diffusion, specifically $(-\Delta)^s-\Delta$ and its parabolic analog with the Marchaud derivative. It introduces generalized weighted average inequalities, narrow region principles, and maximum principles to overcome scaling incompatibilities and to enable sliding in both elliptic and parabolic settings. These tools yield rigorous monotonicity and one-dimensional symmetry results in bounded domains, half-spaces, and the whole space, and extend to time-fractional parabolic problems. As an application, the approach resolves a Gibbons-type conjecture for a class of mixed fractional equations, demonstrating the method's potential for broader mixed-operator problems.

Abstract

We investigate elliptic and parabolic equations involving mixed local and nonlocal operators of the form $(-Δ)^s-Δ$, as well as their parabolic counterparts with both the Marchaud fractional time derivative and the classical first-order derivative. A major difficulty in this setting stems from the coexistence of operators with different nonlocal structures and incompatible scaling properties, which obstruct the direct use of classical sliding methods. To address this issue, we develop a refined sliding method suited to mixed local-nonlocal operators. As key technical ingredients, we establish new generalized weighted average inequalities, narrow region principles, and maximum principles in bounded and unbounded domains. These tools enable us to derive monotonicity and one-dimensional symmetry results for mixed elliptic equations in bounded domains, half-spaces, and the whole space, and to extend the analysis to parabolic equations with mixed time derivatives. As an application, we resolve the Gibbons' conjecture for a class of mixed fractional equations.

On sliding methods for mixed local and nonlocal equations and Gibbons' conjecture

TL;DR

The paper develops a refined sliding framework to study equations with mixed local and nonlocal diffusion, specifically and its parabolic analog with the Marchaud derivative. It introduces generalized weighted average inequalities, narrow region principles, and maximum principles to overcome scaling incompatibilities and to enable sliding in both elliptic and parabolic settings. These tools yield rigorous monotonicity and one-dimensional symmetry results in bounded domains, half-spaces, and the whole space, and extend to time-fractional parabolic problems. As an application, the approach resolves a Gibbons-type conjecture for a class of mixed fractional equations, demonstrating the method's potential for broader mixed-operator problems.

Abstract

We investigate elliptic and parabolic equations involving mixed local and nonlocal operators of the form , as well as their parabolic counterparts with both the Marchaud fractional time derivative and the classical first-order derivative. A major difficulty in this setting stems from the coexistence of operators with different nonlocal structures and incompatible scaling properties, which obstruct the direct use of classical sliding methods. To address this issue, we develop a refined sliding method suited to mixed local-nonlocal operators. As key technical ingredients, we establish new generalized weighted average inequalities, narrow region principles, and maximum principles in bounded and unbounded domains. These tools enable us to derive monotonicity and one-dimensional symmetry results for mixed elliptic equations in bounded domains, half-spaces, and the whole space, and to extend the analysis to parabolic equations with mixed time derivatives. As an application, we resolve the Gibbons' conjecture for a class of mixed fractional equations.
Paper Structure (14 sections, 12 theorems, 380 equations)

This paper contains 14 sections, 12 theorems, 380 equations.

Table of Contents

  1. Introduction
  2. Elliptic equations involving mixed local and nonlocal operators
  3. Parabolic equations involving mixed local and nonlocal operators
  4. Various maximum principles and key lemmas
  5. Maximum principles and key lemmas involving elliptic mixed local and nonlocal operators
  6. Maximum principles and key lemmas involving parabolic mixed local and nonlocal operators
  7. Monotonicity and one-dimensional symmetry for elliptic equations
  8. Monotonicity in bounded domains
  9. Monotonicity in $\mathbb{R}_+^n$
  10. One-dimensional symmetry in $\mathbb{R}^n$
  11. Monotonicity and one-dimensional symmetry for parabolic equations
  12. Monotonicity in $\Omega \times \mathbb R$
  13. Monotonicity in $\mathbb{R}_+^n$
  14. One-dimensional symmetry in $\mathbb{R}^n$

Key Result

Theorem 1.1

(Narrow region principle in bounded domains) Let $\Omega$ be a bounded narrow region in $\mathbb{R}^n$. Assume that $w_\tau \in \mathcal{L}_{2s} \cap C^2(\Omega)$ is upper semi-continuous on $\overline{\Omega}$ and satisfies where $c(x)$ bounded from below in $\Omega$. Let $l_n(\Omega)$ denote the width of $\Omega$ in the $x_n$-direction, and assume that $\Omega$ is narrow in this direction in th

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2
  • Remark 3
  • Theorem 1.6
  • Theorem 1.7
  • ...and 19 more