The Bourgain-Br\'ezis-Mironescu formula in arbitrary bounded domains

TL;DR

This paper extends the Bourgain-Brézis-Mironescu formula to arbitrary bounded domains, analyzing the limit of a modified fractional Sobolev seminorm as the fractional parameter approaches 1, and recovers classical results in extension domains.

Contribution

It generalizes the Bourgain-Brézis-Mironescu formula to arbitrary bounded domains, including non-extension domains, providing a broader understanding of fractional Sobolev seminorm limits.

Findings

Established the limit behavior of a modified fractional Sobolev seminorm in arbitrary bounded domains.

Recovered classical Bourgain-Brézis-Mironescu results in extension domains.

Extended the applicability of the formula beyond classical settings.

Abstract

We obtain a Bourgain-Br\'ezis-Mironescu formula on the limit behaviour of a modified fractional Sobolev seminorm when $s\nearrow 1$, which is valid in arbitrary bounded domains. In the case of extension domains, we recover the classical result.