A Homogenized Maz'ya--Shaposhnikova formula
Abstract
We study the asymptotic behaviour of fractional Gagliardo-type energies with a periodic, bounded and bounded away from zero coefficient depending on both variables, in the joint limit of vanishing fractional exponent s→0+ and vanishing period ε→0+. Assuming without loss of generality that ε−2s converges to some λ∈[1,+∞], we prove that the energies, rescaled by s, Γ-converge in the weak topology of L2(ℝd) to a multiple of the squared L2 norm, whose constant is given by an effective coefficient aλ defined through a cell problem. The coefficient aλ 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 (λ=1) and the arithmetic mean (λ=+∞) 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<∞ is also discussed.
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BibTeX
@misc{braides2026homogenized,
title = {{A Homogenized Maz'ya--Shaposhnikova formula}},
author = {Andrea Braides and Marco Picerni and Alec Jacopo Almo Schiavoni Piazza},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10935},
eprint = {2610.10935},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10935},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Homogenized Maz'ya--Shaposhnikova formula
AU - Andrea Braides
AU - Marco Picerni
AU - Alec Jacopo Almo Schiavoni Piazza
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10935
UR - https://arxiv.org/abs/2610.10935
AB - We study the asymptotic behaviour of fractional Gagliardo-type energies with a periodic, bounded and bounded away from zero coefficient depending on both variables, in the joint limit of vanishing fractional exponent s→0^+ and vanishing period ε→0^+. Assuming without loss of generality that ε^−2s converges to some λ∈[1,+∞], we prove that the energies, rescaled by s, Γ-converge in the weak topology of L^2(ℝ^d) to a multiple of the squared L^2 norm, whose constant is given by an effective coefficient a_λ defined through a cell problem. The coefficient a_λ 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 (λ=1) and the arithmetic mean (λ=+∞) 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<∞ is also discussed.
ER -