Rearrangements and the Maz'ya-Shaposhnikova limit: $\Gamma$-convergence for truncated interactions

Abstract

We give a proof of the $\Gamma$-liminf inequality for the Maz'ya--Shaposhni\-kova 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 ($\lambda=1$).