Characterizations to the fractional Sobolev inequality

TL;DR

This paper provides a characterization of the fractional Sobolev inequality through fractional isocapacitary and isoperimetric inequalities, offering new insights into the relationship between fractional capacity and perimeter.

Contribution

It introduces a new characterization of the fractional Sobolev inequality using fractional isocapacitary and isoperimetric inequalities, including sufficient conditions and examples.

Findings

Fractional capacity of the closure is bounded by fractional perimeter under certain conditions.

Provides a sufficient condition linking fractional capacity and perimeter.

Includes examples illustrating the theoretical results.

Abstract

We characterize the fractional Sobolev inequality with fractional isocapacitary and isoperimetric inequalities. We give a sufficient condition and examples so that the fractional capacity of the closure of an open set is bounded above by the fractional perimeter of its interior.