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

TL;DR

This paper characterizes the limiting behavior of $(s,p)$-Gagliardo energies on the flat torus as the fractional parameter approaches 0 and 1, revealing connections to classical and nonlocal energies.

Contribution

It rigorously establishes the $ ext{Gamma}$-convergence of these energies to classical and nonlocal limits in a periodic setting, extending previous results.

Findings

As $s o 0^+$, the rescaled energy $ ext{s} imes ext{F}_p^s$ $ ext{Gamma}$-converges to a double integral functional.

As $s o 1^-$, the rescaled energy $(1- ext{s}) imes ext{F}_p^s$ $ ext{Gamma}$-converges to the classical Dirichlet $p$-energy.

The minimizer among piecewise affine periodic functions is achieved when jump points are equally spaced.

Abstract

This paper deals with the variational analysis, for every $s \in (0,1)$ and $p \in [1,+\infty)$, 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 $\Gamma$-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 $s\mathcal{F}_p^s$ $\Gamma$-converges to a functional $\mathcal{F}_p^0$ defined by the double integral of $|u(x)-u(y)|^p$ over the periodic cell. Then, for the limit as $s \to 1^-$, we establish that the rescaled energy $(1-s)\mathcal{F}_p^s$ $\Gamma$-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.