Bourgain-Brezis-Mironescu极限中的均匀化与平均化

中文摘要 AI 辅助

我们研究了具有周期系数的二次分数阶能量的均匀化与Bourgain--Brezis--Mironescu极限的同步渐近行为。这些泛函依赖于两个小参数：系数的周期和趋近于局部极限的分数阶指数。我们证明了Γ-极限完全由这些参数的相对尺度决定。在两个极端情形下，恢复了预期的尺度分离，分别得到均匀化的Dirichlet能量或具有平均系数的能量。然而，在临界对数尺度下，两种机制共存，极限是均匀化能量与平均化能量的凸组合，其权重显式依赖于两个渐近过程的相对尺度。证明结合了短程相互作用的均匀化分析与一个爆破论证，后者表明长程相互作用通过Riemann--Lebesgue型论证被简单地平均化。

英文摘要

We study the simultaneous asymptotic behaviour of homogenization and the Bourgain--Brezis--Mironescu limit for quadratic fractional energies with periodic coefficients. The functionals depend on two small parameters: the period of the coefficients and the fractional exponent approaching the local limit. We prove that the $Γ$-limit is completely determined by the relative scaling of these parameters. In the two extreme regimes, one recovers the expected separation of scales, yielding either the homogenized Dirichlet energy or the energy with averaged coefficients. At the critical logarithmic scaling, however, the two mechanisms coexist, and the limit is a convex combination of the homogenized and averaged energies, with weights depending explicitly on the relative scaling of the two asymptotic processes. The proof combines the homogenization analysis of the short-range interactions with a blow-up argument showing that long-range interactions are simply averaged using a Riemann--Lebesgue-type argument.

发表机构

• University of Rome Tor Vergata（罗马托尔维加塔大学）
• SISSA（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。