Rearrangements and the Maz'ya-Shaposhnikova limit: Γ-convergence for truncated interactions
Abstract
We give a proof of the Γ-liminf inequality for the Maz'ya--Shaposhni\-kova limit, as s→0+, of the fractional Sobolev seminorm, based on spherical rearrangements. The key point is that the rearrangements of a weakly convergent sequence are monotone radial profiles, which converge strongly by Helly's theorem; since the energy does not increase under rearrangement (in the limit), this reduces the liminf inequality to a pointwise convergence statement for radially decreasing functions. As an application, we treat truncated energies, in which the interaction is restricted to a ball of radius rs→∞. Assuming rs−2s→λ∈[0,1], we prove that the scaled energies Γ-converge, with respect to weak L2-convergence, to (1−λ)NωN|u|L22, where ωN is the measure of the unit ball of ℝN. This identifies the contribution of the long-range interactions and gives a continuous interpolation between the classical Maz'ya--Shaposhnikova limit (λ=0) and the vanishing-energy regime (λ=1).
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BibTeX
@misc{braides2026rearrangements,
title = {{Rearrangements and the Maz'ya-Shaposhnikova limit: $Γ$-convergence for truncated interactions}},
author = {Andrea Braides and Anneliese Defranceschi and Enrico Vitali},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05499},
eprint = {2610.05499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05499},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rearrangements and the Maz'ya-Shaposhnikova limit: Γ-convergence for truncated interactions
AU - Andrea Braides
AU - Anneliese Defranceschi
AU - Enrico Vitali
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05499
UR - https://arxiv.org/abs/2610.05499
AB - We give a proof of the Γ-liminf inequality for the Maz'ya--Shaposhni\-kova limit, as s→0^+, of the fractional Sobolev seminorm, based on spherical rearrangements. The key point is that the rearrangements of a weakly convergent sequence are monotone radial profiles, which converge strongly by Helly's theorem; since the energy does not increase under rearrangement (in the limit), this reduces the liminf inequality to a pointwise convergence statement for radially decreasing functions. As an application, we treat truncated energies, in which the interaction is restricted to a ball of radius r_s→∞. Assuming r_s^−2s→λ∈[0,1], we prove that the scaled energies Γ-converge, with respect to weak L^2-convergence, to (1−λ)Nω_N|u|_L^2^2, where ω_N is the measure of the unit ball of ℝ^N. This identifies the contribution of the long-range interactions and gives a continuous interpolation between the classical Maz'ya--Shaposhnikova limit (λ=0) and the vanishing-energy regime (λ=1).
ER -