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Asymptotics of variational eigenvalues for a general nonlocal $p$-Laplacian with varying horizon

Guillermo García-Sáez

TL;DR

The paper develops a general nonlocal p-Laplacian built from a radial kernel ρ with finite horizon and studies its Dirichlet eigenproblem via a robust variational framework. It establishes existence of variational eigenvalues and eigenfunctions in both the Hilbertian case (p=2) and the nonlinear case (p≠2) using min-max principles and Lusternik-Schnirelman theory, extended to general kernels. A central contribution is the Γ-convergence analysis with varying horizon δ, showing that δ→0^+ recovers local p-Laplacian eigenstructures while δ→∞ yields the H^{s,p}-Laplacian spectrum, along with convergence of eigenfunctions. The results connect nonlocal models to classical PDE and fractional models, providing stability and asymptotic consistency for applications in peridynamics and nonlocal diffusion.

Abstract

From the recent developing of nonlocal gradients with finite horizon $δ>0$ based on general kernels, we introduce a new nonlocal $p$-Laplacian and study the eigenvalue problem associated with it. Furthermore, by virtue of $Γ$-convergence arguments, we establish stability results of the solutions for varying horizon in the extreme cases $δ\to 0^+$ and $δ\to\infty$, recovering the solutions for the local eigenvalue problem associated with the $p$-Laplacian, and the ones associated with the $H^{s,p}$-Laplacian, respectively.

Asymptotics of variational eigenvalues for a general nonlocal $p$-Laplacian with varying horizon

TL;DR

The paper develops a general nonlocal p-Laplacian built from a radial kernel ρ with finite horizon and studies its Dirichlet eigenproblem via a robust variational framework. It establishes existence of variational eigenvalues and eigenfunctions in both the Hilbertian case (p=2) and the nonlinear case (p≠2) using min-max principles and Lusternik-Schnirelman theory, extended to general kernels. A central contribution is the Γ-convergence analysis with varying horizon δ, showing that δ→0^+ recovers local p-Laplacian eigenstructures while δ→∞ yields the H^{s,p}-Laplacian spectrum, along with convergence of eigenfunctions. The results connect nonlocal models to classical PDE and fractional models, providing stability and asymptotic consistency for applications in peridynamics and nonlocal diffusion.

Abstract

From the recent developing of nonlocal gradients with finite horizon based on general kernels, we introduce a new nonlocal -Laplacian and study the eigenvalue problem associated with it. Furthermore, by virtue of -convergence arguments, we establish stability results of the solutions for varying horizon in the extreme cases and , recovering the solutions for the local eigenvalue problem associated with the -Laplacian, and the ones associated with the -Laplacian, respectively.
Paper Structure (13 sections, 27 theorems, 232 equations)

This paper contains 13 sections, 27 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Nonlocal gradients
  5. Nonlocal Sobolev spaces
  6. Scaled Kernels
  7. Krasnoselskii genus
  8. Abstract minmax results
  9. The nonlocal $p$-Laplacian
  10. The eigenvalue problem for the $\rho$-Laplacian
  11. The Hilbertian case
  12. The nonlinear case
  13. $\Gamma$-convergence of the eigenvalue problem for varying horizon

Key Result

Lemma 3.3

Let $u\in C_c^\infty({\mathbb R}^n)$ and $v\in C_c^\infty({\mathbb R}^n;{\mathbb R}^n)$. Then,

Theorems & Definitions (33)

  • Definition 3.1: Nonlocal gradient
  • Definition 3.2
  • Lemma 3.3: Integration by parts
  • Lemma 3.4: Translation operators
  • Theorem 3.5
  • Theorem 3.6
  • Lemma 3.7: Localization for vanishing horizon
  • Proposition 3.8
  • Definition 3.9: Kranoselskii genus
  • Lemma 3.10
  • ...and 23 more