Submitted Paper

Inserted: 8 oct 2026
Last Updated: 8 oct 2026

Year: 2026

Abstract:

We study the asymptotic behavior of fractional Gagliardo-type energies with a periodic, bounded, and bounded away from zero coefficient depending on both variables, in the joint limit of the vanishing fractional exponent $s\to0^+$ and the vanishing period $\varepsilon\to0^+$. Assuming that $\varepsilon^{-2s}$ converges to some $\lambda\in[1,+\infty]$, which holds up to subsequences, we prove that the energies, rescaled by $s$, $\Gamma$-converge in the weak topology of $L^2(\mathbb{R}^d)$ to a multiple of the square of the $L^2$ norm, whose constant is given by an effective coefficient $a_\lambda$ defined through a cell problem. The coefficient $a_\lambda$ results from the competition between the averaging of long-range interactions, which drives the Maz'ya-Shaposhnikova limit, and the nonlocal energy of microscopic oscillations at finite range. It interpolates between the harmonic mean ($\lambda=1$) and the arithmetic mean ($\lambda=+\infty$) of a coefficient obtained by averaging interactions "at infinity''. The proof relies on Riemann-Lebesgue-type lemmas for singular kernels, on a blow-up argument for the lower bound, and on the logarithmic-scaling properties of the energies. The case $1<p<\infty$ is also discussed.

Keywords: Gamma-convergence, fractional Sobolev spaces, Gagliardo seminorms, Maz'ya-Shaposhnikova formula, nonlocal homogenization

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