$Γ-$convergence of energy functionals in fractional Orlicz spaces beyond the $Δ_2$ condition
Ignacio Ceresa Dussel, Julián Fernández Bonder, Ariel Salort
TL;DR
This work extends the $\Gamma$-convergence analysis of nonlocal energy functionals to fractional Orlicz spaces without the $\Delta_2$ growth assumption. It proves a liminf inequality for sequences with $s_k\to1$ and $u_k\to u$ in $E^A(\mathbb{R}^n)$, establishing the $\Gamma$-convergence of $\mathcal{J}_s$ to a local limit functional $\mathcal{J}$ that captures the asymptotic behavior of fractional Orlicz-Sobolev spaces. The main technical contribution is deriving the liminf inequality without $\Delta_2$ by regularizing and leveraging uniform convergence on bounded sets, together with a peridynamic extension. The results broaden applicability to general Orlicz growth conditions and clarify limits for peridynamic-type energies in the absence of $\Delta_2$, highlighting the role of the complementary function and Matuszewska-type limits in the limit process.
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 $Δ_2$ 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 $Γ-$convergence of the functional $\mathcal{J}_s$. Finally, we extend our result to the study of the so called \emph{fractional peridynamic} case.