Fractional order Orlicz-Sobolev spaces

TL;DR

This paper introduces fractional order Orlicz-Sobolev spaces, demonstrating their convergence to classical spaces as the fractional parameter approaches one, and explores implications like gamma-convergence and solution convergence for fractional operators.

Contribution

It defines fractional Orlicz-Sobolev spaces and proves their convergence to classical spaces, extending Bourgain-Brezis-Mironescu results to this new setting.

Findings

Convergence of fractional Orlicz-Sobolev spaces to classical spaces as s approaches 1

Gamma-convergence of modulars in the fractional setting

Convergence of solutions for fractional Δ_g operators

Abstract

In this paper we define the fractional order Orlicz-Sobolev spaces, and prove its convergence to the classical Orlicz-Sobolev spaces when the fractional parameter $s\uparrow 1$ in the spirit of the celebrated result of Bourgain-Brezis-Mironescu. We then deduce some consequences such as $\Gamma-$convergence of the modulars and convergence of solutions for some fractional versions of the $\Delta_g$ operator as the fractional parameter $s\uparrow 1$.