A Bourgain-Brezis-Mironescu formula for anisotropic fractional Sobolev spaces and applications to anisotropic fractional differential equations

TL;DR

This paper extends Bourgain-Brezis-Mironescu type formulas to anisotropic fractional Sobolev spaces, providing new tools for analyzing anisotropic fractional p-Laplacian equations and their solution stability.

Contribution

It introduces anisotropic fractional Sobolev space formulas and applies them to stability analysis of anisotropic fractional p-Laplacian equations, advancing the understanding of these operators.

Findings

Established Bourgain-Brezis-Mironescu formulas for anisotropic fractional energies

Derived Maz'ya-Shaposhnikova type results for these energies

Applied results to analyze solution stability of anisotropic fractional p-Laplacian equations

Abstract

In this paper we prove Bourgain-Brezis-Mironescu's type results (cf. \cite{BBM2001}) (BBM for short) for an energy functional which is strongly related to the fractional anisotropic p-Laplacian. We also provide with the analogous of Maz'ya-Shaposhnikova (see \cite{MS}) type results for these energies and finally we apply these results to analyze the stability of solutions to anisotropic fractional $p-$laplacian equations.