$\mathrm{\Gamma }-$convergence of energy functionals in fractional Orlicz spaces beyond the ${\mathrm{\Delta }}_2$ condition

Given a Young function $A$, $n\ge 1$ and $s\in (0,1)$ we consider the energy functional $${\mathcal{J}}_s(u)=(1-s){\iint }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}A\left(\frac{\mathrm{\mid }u(x)-u(y)\mathrm{\mid }}{\mathrm{\mid }x-y{\mathrm{\mid }}^s}\right)\frac{dxdy}{\mathrm{\mid }x-y{\mathrm{\mid }}^n}\mathrm{.}$$ Without assuming the ${\mathrm{\Delta }}_2$ 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 $$\mathcal{J}(u)\le \underset{k\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}{\mathcal{J}}_{s_k}(u_k),$$ 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 $\mathrm{\Gamma }-$convergence of the functional ${\mathcal{J}}_s$. Finally, we extend our result to the study of the so called \emph{fractional peridynamic} case.