发表机构
University of Rome Tor Vergata; SISSA（罗马托尔维加塔大学; 的里雅斯特国际高等研究学校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究双变量依赖的周期系数下分数阶Gagliardo型能量在s→0+和ε→0+联合极限的渐近行为，证明经s缩放的能量Γ收敛到L²范数平方的倍数，该倍数由插值调和平均与算术平均的有效系数a_λ给出，方法含奇异核引理等，还讨论了1<p<∞情形。
AI 中文摘要
我们研究依赖于双变量的周期、有界且远离零的系数下，分数阶 Gagliardo 型能量在分数阶指数 $s\to0^+$ 和周期 $\varepsilon\to0^+$ 同时消失时的渐近行为。不失一般性假设 $\varepsilon^{-2s}$ 收敛到某个 $\lambda\in[1,+\text{\infty}]$，我们证明：经 $s$ 缩放后的能量在 $L^2(\mathbb{R}^d)$ 的弱拓扑下 $\Gamma$ 收敛到 $L^2$ 范数平方的某个倍数，该倍数由单元问题定义的有效系数 $a_\lambda$ 给出。系数 $a_\lambda$ 源于长程相互作用平均（驱动 Maz'ya--Shaposhnikova 极限）与有限范围微观振荡的非局部能量之间的竞争，它插值于“无穷远处”相互作用平均得到的系数的调和平均（$\lambda=1$）和算术平均（$\lambda=+\text{\infty}$）之间。证明依赖于奇异核的 Riemann--Lebesgue 型引理、下界的爆破论证以及能量的对数标度性质，同时也讨论了 $1<p<\infty$ 的情形。
英文摘要
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 $s\to0^+$ and vanishing period $\varepsilon\to0^+$. Assuming without loss of generality that $\varepsilon^{-2s}$ converges to some $λ\in[1,+\infty]$, we prove that the energies, rescaled by $s$, $Γ$-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 $a_λ$ 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 ($λ=+\infty$) 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.
