标题： 一种分数薄层的Bourgain-Brezis-Mironescu结果 显示英文标题

标题： A Bourgain-Brezis-Mironescu result for fractional thin films

摘要： We consider the limit of squared $H^s$-Gagliardo seminorms on thin domains of the form $\Omega_\varepsilon=\omega\times(0,\varepsilon)$ in $\mathbb R^d$. 当$\varepsilon$固定时，通过乘以$1-s$这样的半范数已被证明当$s\to 1^-$时收敛到一个与维度有关的常数$c_d$乘以$\Omega_\varepsilon$上的 Dirichlet 积分，这是由 Bourgain、Brezis 和 Mironescu 证明的。 在另一方面，这样的狄利克雷积分除以$\varepsilon$随着$\varepsilon\to 0$收敛到$\omega$上的一个维数降低的狄利克雷积分。 我们证明，如果我们同时让$\varepsilon\to 0$和$s\to 1$，则这些平方半范数在乘以$(1-s) \varepsilon^{2s-3}$时，仍然收敛到相同的维数约简极限，而与$s$和$\varepsilon$的相对收敛速度无关。 这个系数结合了几何缩放$\varepsilon^{-1}$和有关$H^s$-Gagliardo 无 norms 的相关相互作用是在尺度$\varepsilon$的事实。 我们还研究了通过乘以$(1-s)\varepsilon^{-1}$得到的常规膜尺度，这突出了{\em 临界标度}$1-s\sim|\log\varepsilon|^{-1}$，以及当$\varepsilon\to 0$固定时的极限$s$。 显示英文摘要

摘要： We consider the limit of squared $H^s$-Gagliardo seminorms on thin domains of the form $\Omega_\varepsilon=\omega\times(0,\varepsilon)$ in $\mathbb R^d$. When $\varepsilon$ is fixed, multiplying by $1-s$ such seminorms have been proved to converge as $s\to 1^-$ to a dimensional constant $c_d$ times the Dirichlet integral on $\Omega_\varepsilon$ by Bourgain, Brezis and Mironescu. In its turn such Dirichlet integrals divided by $\varepsilon$ converge as $\varepsilon\to 0$ to a dimensionally reduced Dirichlet integral on $\omega$. We prove that if we let simultaneously $\varepsilon\to 0$ and $s\to 1$ then these squared seminorms still converge to the same dimensionally reduced limit when multiplied by $(1-s) \varepsilon^{2s-3}$, independently of the relative converge speed of $s$ and $\varepsilon$. This coefficient combines the geometrical scaling $\varepsilon^{-1}$ and the fact that relevant interactions for the $H^s$-Gagliardo seminorms are those at scale $\varepsilon$. We also study the usual membrane scaling, obtained by multiplying by $(1-s)\varepsilon^{-1}$, which highlighs the {\em critical scaling} $1-s\sim|\log\varepsilon|^{-1}$, and the limit when $\varepsilon\to 0$ at fixed $s$.