Asymptotic behavior for anisotropic fractional energies

TL;DR

This paper studies the asymptotic behavior of anisotropic fractional energies as the fractional parameter approaches 0 and 1, analyzing minimizers and homogenization effects in these limits.

Contribution

It extends the analysis of fractional energies to anisotropic cases and explores their asymptotic behavior, including homogenization phenomena as the parameter approaches 1.

Findings

Asymptotic behavior characterized for s approaching 0 and 1

Analysis of minimizers in the s1 limit

Identification of homogenization effects in combined localization phenomena

Abstract

In this paper we investigate the asymptotic behavior of anisotropic fractional energies as the fractional parameter $s\in (0,1)$ approaches both $s\uparrow 1$ and $s\downarrow 0$ in the spirit of the celebrated papers of Bourgain-Brezis-Mironescu \cite{BBM} and Maz'ya-Shaposhnikova \cite{MS}. Then, focusing con the case $s\uparrow 1$ we analyze the behavior of solutions to the corresponding minimization problems and finally, we also study the problem where a homogenization effect is combined with the localization phenomena that occurs when $s\uparrow 1$.