The first non-zero Neumann $p-$fractional eigenvalue

TL;DR

This paper investigates the asymptotic behavior of the first non-zero Neumann p-fractional eigenvalue as the fractional parameter approaches 1 and as p tends to infinity, revealing connections to classical and H"older Laplacians.

Contribution

It establishes the limiting behavior of the eigenvalues in two regimes, linking fractional eigenvalues to classical Laplacians and H"older infinity-Laplacian eigenvalues.

Findings

$(1-s)\lambda_1(s,p)$ converges to the classical Neumann eigenvalue of the p-Laplacian as $s\to 1^-$.

$\lambda_1(1,s)^{1/p}$ converges to an eigenvalue of the H"older $\infty$-Laplacian as $p\to\infty$.

Existence of a constant $\mathcal{K}$ relating fractional and classical eigenvalues.

Abstract

In this work we study the asymptotic behavior of the first non-zero Neumann $p-$fractional eigenvalue $\lambda_1(s,p)$ as $s\to 1^-$ and as $p\to\infty.$ We show that there exists a constant $\mathcal{K}$ such that $\mathcal{K}(1-s)\lambda_1(s,p)$ goes to the first non-zero Neumann eigenvalue of the $p-$Laplacian. While in the limit case $p\to \infty,$ we prove that $\lambda_1(1,s)^{1/p}$ goes to an eigenvalue of the H\"older $\infty-$Laplacian.