$\Gamma-$convergence of energy functionals in fractional Orlicz spaces beyond the $\Delta_2$ condition

TL;DR

This paper establishes the $ ext{Gamma}$-convergence of fractional Orlicz energy functionals as the fractional parameter approaches 1, without requiring the $ ext{Delta}_2$ condition, and extends results to fractional peridynamics.

Contribution

It proves the liminf inequality and $ ext{Gamma}$-convergence of fractional Orlicz energy functionals without the $ ext{Delta}_2$ condition, extending to fractional peridynamics.

Findings

Proved liminf inequality for fractional Orlicz energies as s approaches 1.

Established $ ext{Gamma}$-convergence of these functionals.

Extended results to fractional peridynamic models.

Abstract

Given a Young function $A$, $n\geq 1$ and $s\in(0,1)$ we consider the energy functional $$ \mathcal{J}_s(u)=(1-s)\iint_{\mathbb{R}^n\times \mathbb{R}^n} A\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^n}. $$ Without assuming the $\Delta_2$ condition on $A$ not its conjugated function $\bar A$, we prove the following liminf inequality: if $u\in E^A(\mathbb{R}^n)$ and $\{u_k\}_{k\in\mathbb{N}}\subset E^A(\mathbb{R}^n)$ is such that $u_k\to u$ in $E^A(\mathbb{R}^n)$, and $s_k\to 1$, then $$ \mathcal{J}(u) \leq \liminf_{k\to\infty } \mathcal{J}_{s_k}(u_k), $$ where $\mathcal{J}$ is a limit functional related with the behavior of the fractional Orlicz-Sobolev spaces as $s\to 1^+$. As a direct consequence, we obtain the $\Gamma-$convergence of the functional $\mathcal{J}_s$. Finally, we extend our result to the study of the so called \emph{fractional peridynamic} case.