Preprint

Inserted: 4 oct 2026
Last Updated: 4 oct 2026

Year: 2026

Abstract:

We give a proof of the $\Gamma$-liminf inequality for the Maz'ya-Shaposhnikova limit, as $s\to0^+$, 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 $r_s\to\infty$. Assuming $r_s^{-2s}\to\lambda\in[0,1]$, we prove that the scaled energies $\Gamma$-converge, with respect to weak $L^2$-convergence, to $(1-\lambda)N\omega_N \|u\|_{L^2}^2$, where $\omega_N$ is the measure of the unit ball of $\mathbb R^N$. This identifies the contribution of the long-range interactions and gives a continuous interpolation between the classical Maz'ya--Shaposhnikova limit ($\lambda=0$) and the vanishing-energy regime ($\lambda=1$).

Keywords: fractional Sobolev spaces, Symmetrization, Maz’ya–Shaposhnikova theorem, spherical rearrange- ment

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