Spectral stability for the perydinamic fractional $p$-Laplacian

TL;DR

This paper studies how the spectrum of the peridynamic fractional p-Laplacian operator changes as the interaction horizon parameter varies, showing convergence to classical and fractional p-Laplacians in different limits.

Contribution

It establishes spectral convergence results for the peridynamic fractional p-Laplacian as the horizon parameter approaches zero or infinity, connecting nonlocal and local operators.

Findings

Spectral convergence to classical p-Laplacian as δ→0+

Spectral convergence to fractional p-Laplacian as δ→+∞

Provides a rigorous analysis of spectral stability in peridynamic models

Abstract

In this work we analyze the behavior of the spectrum of the peridynamic fractional $p$-Laplacian, $(-\Delta_p)_{\delta}^s$, under the limit process $\delta\to0^+$ or $\delta\to+\infty$. We prove spectral convergence to the classical $p$-Laplacian under a suitable scaling as $\delta\to0^+$ and to the fractional $p$-Laplacian as $\delta\to+\infty$.