$\mathrm{\Gamma }$-convergence, variational analysis and characterisation of minimisers for $(s,p)$-Gagliardo energies in the flat $d$-torus

This paper deals with the variational analysis, for every $s\in (0,1)$ and $p\in [1,+\mathrm{\infty })$, of $(s,p)$-Gagliardo seminorms in a periodic setting. First, we consider the space of $L^p$, $T$-periodic functions and define the energy functional ${\mathcal{F}}_p^s$ as the density of the $d$-dimensional $(s,p)$-Gagliardo seminorm over the periodic cell. Our goal is to rigorously characterise the $\mathrm{\Gamma }$-limits of this functional as the fractional parameter $s$ approaches its endpoint values, $0^{+}$ and $1^{-}$. We prove that, as $s\to 0^{+}$, the rescaled energy $s{\mathcal{F}}_p^s$ $\mathrm{\Gamma }$-converges to a functional ${\mathcal{F}}_p^0$ defined by the double integral of $\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^p$ over the periodic cell. Then, for the limit as $s\to 1^{-}$, we establish that the rescaled energy $(1-s){\mathcal{F}}_p^s$ $\mathrm{\Gamma }$-converges to the classical Dirichlet $p$-energy, extending known results from bounded domains to the periodic framework. Finally, we analyse the one-dimensional minimiser of the energy ${\mathcal{F}}_p^s$ for $s\in (0,1)$ and the limit functional ${\mathcal{F}}_p^0$ within the special class of piecewise affine periodic functions whose distributional derivative consists of a constant absolutely continuous part and a singular part with opposite sign and quantised jumps. In this setting, the energy depends only on the position of these jump points, and we prove that the absolute minimiser is achieved by their equispaced configuration.