Alejandro Ortega, Pervasiveness of the p-Laplace operator under localization of fractional g-Laplace operators

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DOI: 10.23952/jnva.9.2025.3.04

Volume 9, Issue 3, 1 June 2025, Pages 373-395

Abstract. In this paper, we analyze the behavior of the truncated functionals as

$\int_{\mathbb{R}^N}\int_{B(x,\delta)} G\left(\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)\frac{dydx}{|x-y|^N}$ for $\delta\to0^+,$

where $G$ is an Orlicz function which is assumed to be regularly varying at $0$ . A prototype of such function is given by $G(t)=t^p(1+|\log(t)|)$ with $p\geq2$ . These kinds of functionals arise naturally in peridynamics, where long-range interactions are neglected and only those that exerted at distance smaller than $\delta>0$ are taken into account, i.e., the horizon $\delta>0$ represents the range of interactions or nonlocality. This paper is inspired by the celebrated result by Bourgain, Brezis and Mironescu, who analyzed the limit $s\to1^-$ with $G(t)=t^p$ . In particular, we prove that, under appropriate conditions,

$\lim\limits_{\delta\to0^+}\frac{p(1-s)}{G(\delta^{1-s})}\int_{\mathbb{R}^N}\int_{B(x,\delta)}G\left(\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)\frac{dydx}{|x-y|^N}=K_{N,p}\int_{\mathbb{R}^N}|\nabla u(x)|^p dx,$

for $p=index(G)$ and an explicit constant $K_{N,p}>0$ . Moreover, the converse is also true if the above localization limit exist as $\delta\to 0^+$ , and the Orlicz function $G$ is a regularly varying function with $index(G)=p$ .

How to Cite this Article:
A. Ortega, Pervasiveness of the p-Laplace operator under localization of fractional g-Laplace operators, J. Nonlinear Var. Anal. 9 (2025), 373-395.