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The Neumann condition for the superposition of fractional Laplacians

Serena Dipierro, Edoardo Proietti Lippi, Caterina Sportelli, Enrico Valdinoci

TL;DR

The paper develops a unified variational framework for Neumann conditions tied to the operator $L_{\alpha,\mu}(u)=-\alpha\Delta u+\int_{(0,1)}(-\Delta)^s u\,d\mu(s)$, allowing finite or infinite superpositions of fractional Laplacians. By leveraging an energy-based fractional normal derivative $\mathscr{N}_s$, it derives integration-by-parts formulas, proves existence and uniqueness of weak solutions (up to constants), and establishes a spectral theory with a complete eigenbasis, as well as heat-flow and long-time behavior results. The authors also prove a minimization property linking homogeneous $(\alpha,\mu)$-Neumann data to energy minimizers, study continuity across the boundary, and connect the framework to a superposition of fractional perimeters via a normalized Neumann derivative. The framework covers mixed local-nonlocal operators and both discrete and continuous mixtures of orders, offering a versatile tool for modeling nonlocal diffusion and long-range interactions with multiple scales. These contributions advance nonlocal Neumann theory and provide a bridge between variational methods, spectral analysis, and geometric interpretations of perimeters in superposed fractional settings.

Abstract

We present a new functional setting for Neumann conditions related to the superposition of (possibly infinitely many) fractional Laplace operators. We will introduce some bespoke functional framework and present minimization properties, existence and uniqueness results, asymptotic formulas, spectral analyses, rigidity results, integration by parts formulas, superpositions of fractional perimeters, as well as a study of the associated heat equation.

The Neumann condition for the superposition of fractional Laplacians

TL;DR

The paper develops a unified variational framework for Neumann conditions tied to the operator , allowing finite or infinite superpositions of fractional Laplacians. By leveraging an energy-based fractional normal derivative , it derives integration-by-parts formulas, proves existence and uniqueness of weak solutions (up to constants), and establishes a spectral theory with a complete eigenbasis, as well as heat-flow and long-time behavior results. The authors also prove a minimization property linking homogeneous -Neumann data to energy minimizers, study continuity across the boundary, and connect the framework to a superposition of fractional perimeters via a normalized Neumann derivative. The framework covers mixed local-nonlocal operators and both discrete and continuous mixtures of orders, offering a versatile tool for modeling nonlocal diffusion and long-range interactions with multiple scales. These contributions advance nonlocal Neumann theory and provide a bridge between variational methods, spectral analysis, and geometric interpretations of perimeters in superposed fractional settings.

Abstract

We present a new functional setting for Neumann conditions related to the superposition of (possibly infinitely many) fractional Laplace operators. We will introduce some bespoke functional framework and present minimization properties, existence and uniqueness results, asymptotic formulas, spectral analyses, rigidity results, integration by parts formulas, superpositions of fractional perimeters, as well as a study of the associated heat equation.
Paper Structure (17 sections, 20 theorems, 250 equations, 2 figures)

This paper contains 17 sections, 20 theorems, 250 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Superposition of fractional Laplacians and Neumann conditions
  3. Main results
  4. Examples and possible applications
  5. Organization of the paper
  6. Functional setting
  7. Integration by parts formulas and minimization property
  8. Weak solutions with ($\alpha,\mu$)-Neumann conditions
  9. Eigenvalues and eigenfunctions
  10. The heat equation
  11. Continuity properties
  12. The superposition of fractional perimeters
  13. Proof of Theorem \ref{['uptoconstant']}
  14. Proof of \ref{['Hmuhilbert']}
  15. Proof of Proposition \ref{['propfunctional']}
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $u:\mathbb{R}^N\to \mathbb{R}$ with $u\in L^1(\Omega)$. For all $x\in \mathbb{R}^N\setminus \overline{\Omega}$, we define We set where $E_1$ stands for $E_u$ when $u\equiv 1$. Then,

Figures (2)

  • Figure 1: The rigid motion $\mathcal{R}_k$ of $\Omega$.
  • Figure 2: The set $\mathcal{C}$ (in blue).

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 34 more