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Maximum principles and spectral analysis for the superposition of operators of fractional order

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

TL;DR

This work studies the nonlocal superposition operator $L_\mu u=\int_{[0,1]}(-\Delta)^s u\,d\mu(s)$ with a signed measure $\mu=\mu^+-\mu^-$, motivated by mixtures of diffusion patterns in anomalous diffusion and population dynamics. It develops a rigorous variational framework via spaces $X(\mathbb{R}^N)$ and $X(\Omega)$ and shows how a small negative part $\mu^-$ can be absorbed into the positive part under a Sobolev-analytic regime, yielding Sobolev embeddings and norm equivalences. The authors establish weak and strong maximum principles for the positive part $L^+_\mu$, while demonstrating that the full operator $L_\mu$ may fail to satisfy a maximum principle in the presence of sign-changing components, with a concrete counterexample. They then complete the spectral theory by constructing a diverging sequence of Dirichlet eigenvalues $\lambda_k$ with finite multiplicities, proving that eigenfunctions form an orthonormal basis of $L^2(\Omega)$ and an orthogonal basis of $X(\Omega)$, and providing a variational scheme to compute higher eigenvalues. This yields a comprehensive framework for analyzing mixed-order, possibly sign-changing nonlocal operators and extends fractional Laplacian spectral theory to superposed, signed-measure settings.

Abstract

We consider a "superposition operator" obtained through the continuous superposition of operators of mixed fractional order, modulated by a signed Borel finite measure defined over the set $[0, 1]$. The relevance of this operator is rooted in the fact that it incorporates special and significant cases of interest, like the mixed operator $-Δ+ (-Δ)^s$, the (possibly) infinite sum of fractional Laplacians and allows to consider operators carrying a "wrong sign". We first outline weak and strong maximum principles for this type of operators. Then, we complete the spectral analysis for the related Dirichlet eigenvalue problem started in [DPLSV25b].

Maximum principles and spectral analysis for the superposition of operators of fractional order

TL;DR

This work studies the nonlocal superposition operator with a signed measure , motivated by mixtures of diffusion patterns in anomalous diffusion and population dynamics. It develops a rigorous variational framework via spaces and and shows how a small negative part can be absorbed into the positive part under a Sobolev-analytic regime, yielding Sobolev embeddings and norm equivalences. The authors establish weak and strong maximum principles for the positive part , while demonstrating that the full operator may fail to satisfy a maximum principle in the presence of sign-changing components, with a concrete counterexample. They then complete the spectral theory by constructing a diverging sequence of Dirichlet eigenvalues with finite multiplicities, proving that eigenfunctions form an orthonormal basis of and an orthogonal basis of , and providing a variational scheme to compute higher eigenvalues. This yields a comprehensive framework for analyzing mixed-order, possibly sign-changing nonlocal operators and extends fractional Laplacian spectral theory to superposed, signed-measure settings.

Abstract

We consider a "superposition operator" obtained through the continuous superposition of operators of mixed fractional order, modulated by a signed Borel finite measure defined over the set . The relevance of this operator is rooted in the fact that it incorporates special and significant cases of interest, like the mixed operator , the (possibly) infinite sum of fractional Laplacians and allows to consider operators carrying a "wrong sign". We first outline weak and strong maximum principles for this type of operators. Then, we complete the spectral analysis for the related Dirichlet eigenvalue problem started in [DPLSV25b].

Paper Structure

This paper contains 12 sections, 15 theorems, 161 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Why superposition operators?
  3. The mathematical setting
  4. Maximum principles
  5. Failure of the maximum principle
  6. Eigenvalues and eigenfunctions
  7. Organization of the paper
  8. Preliminary results
  9. Maximum principles
  10. The eigenvalue problem
  11. Proof of \ref{['counterexamplemaxprinciple']}
  12. Proof of \ref{['baseX']}

Key Result

Theorem 1.1

Let $\Omega$ be an open and bounded subset of $\mathbb{R}^N$ with Lipschitz boundary. Let $\mu$ satisfy mu0. Let $u\in X(\mathbb{R}^N)$ be such that $L^+_\mu u\geqslant 0$ in $\Omega$ in the weak sense. SupposeWith an abuse of notation, this has to be interpreted in the classical trace sense $u\geqs

Figures (1)

  • Figure 1: Portrait of Sandro by Luigi Serafini.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • Definition 3.1
  • ...and 17 more