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 $s\to 0^{+}$ and vanishing period $\epsilon \to 0^{+}$. Assuming without loss of generality that ${\epsilon }^{-2s}$ converges to some $\lambda \in [1,+\infty ]$, 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 squared $L^2$ 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.