Irene Drelichman, Ricardo G. Durán
We obtain a Bourgain-Brézis-Mironescu formula on the limit behaviour of a modified fractional Sobolev seminorm when $s\nearrow 1$, which is valid in arbitrary bounded domains. In the case of extension domains, we recover the classical result.
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We prove a Brezis-Bourgain-Mironescu type formula for a class of nonlocal magnetic spaces, which builds a bridge between a fractional magnetic operator recently introduced and the classical theory.
We prove a general magnetic Bourgain-Brezis-Mironescu formula. In particular, after developing a theory of magnetic bounded variation functions, we prove the validity of the formula in this class.
In this paper we prove Bourgain-Brezis-Mironescu's type results (cf. \cite{BBM2001}) (BBM for short) for an energy functional which is strongly related to the fractional anisotropic p-Laplacian. We also provide with the analogous of Maz'ya-Shaposhnikova (see \cite{MS}) type results for these energies and finally we apply these results to analyze the stability of solutions to anisotropic fractional $p-$laplacian equations.
In this note we prove the following theorem in any Carnot group of step two $\mathbb{G}$: \[ \underset{s\nearrow 1/2}{\lim} (1 - 2s) \mathfrak P_{H,s}(E) = \frac{4}{\sqrt π}\ \mathfrak P_H(E). \] Here, $\mathfrak P_H(E)$ represents the horizontal perimeter of a measurable set $E\subset \mathbb{G}$, whereas the nonlocal horizontal perimeter $\mathfrak P_{H,s}(E)$ is a heat based Besov seminorm. This result represents a dimensionless sub-Riemannian counterpart of a famous characterisation of Bourgain-Brezis-Mironescu and Dávila.
Following the seminal paper by Bourgain, Brezis and Mironescu, we focus on the asymptotic behavior of some nonlocal functionals that, for each $u\in L^2(\mathbb{R}^N)$, are defined as the double integrals of weighted, squared difference quotients of $u$. Given a family of weights $\{ρ_ε\}$, $ε\in (0,1)$, we devise sufficient and necessary conditions on $\{ρ_ε\}$ for the associated nonlocal functionals to converge as $ε\to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.
We establish a non-local integral difference quotient representation for symmetric gradient semi-norms in $BD(Ω)$ and $LD(Ω)$, which does not require the manipulation of distributional derivatives. Our representation extends the formulas for the symmetric gradient established by Mengesha for vector-fields in $W^{1,p}(Ω;\mathbb R^d)$, which are inspired by the gradient semi-norm formulas introduced by Bourgain, Brezis and Mironescu in $W^{1,p}(Ω)$ and by Dávila in $BV(Ω)$.
A Bourgain-Brezis-Mironescu formula accounting for nonlocal antisymmetric exchange interactions
本文推广 Bourgain-Brezis-Mironescu 公式，证明含对称与反对称交换的非局部能量泛函在短程极限下 Γ-收敛到局部泛函，并严格论证手性磁体中的 Dzyaloshinskii-Moriya 相互作用。
Bourgain-Brezis-Mironescu formula for $W^{s,p}_q$-spaces in arbitrary domains
本文证明 Bourgain-Brezis-Mironescu 公式在任意区域中对广义分数阶 Sobolev 空间 $W^{s,p}_q$ 的半范数成立，解决了 Brazke-Schikorra-Yung 提出的公开问题。
A nonlinear superposition operator $T_g$ related to a Borel measurable function $g:\ {\mathbb C}\to {\mathbb C}$ is defined via $T_g(f):=g\circ f$ for any complex-valued function $f$ on ${\mathbb R^n}$. This article is devoted to investigating the mapping properties of $T_g$ on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083-2101], as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion result and the continuity property of $T_g$ on these spaces are obtained.
We introduce a large class of concentrated $p$-Lévy integrable functions approximating the unity, which serves as the core tool from which we provide a nonlocal characterization of Sobolev spaces and the space of functions of bounded variation via nonlocal energies forms. It turns out that this nonlocal characterization is a necessary and sufficient criterion to define Sobolev spaces on domains satisfying the extension property. We also examine the general case where the extension property does not necessarily hold. In the latter case, we establish weak convergence of the nonlocal Radon measur…
We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{s,p} = F^{s}_{p,p}$, and $H^{1,p} = F^{1}_{p,2}$. When $s\to 1$, the $F^{s}_{p,p}$ norm becomes the $F^{1}_{p,p}$ norm but BBM showed that the $W^{s,p}$ norm becomes the $H^{1,p} = F^{1}_{p,2}$ norm. Naively, for $p \neq 2$ this seems like a contradiction, but we resolve this by providing embeddings of $W^{s,p}$ into $F^{s}_{p,q}$ for $q \in \{p,2\}$ with sharp constants with respect to $s \in (0,1)$. As a consequence we obtain an $\mathbb{R}^N$-version of the B…
Bourgain-Brezis-Mironescu formula for Riesz Potentials
研究Riesz势的非局部势算子在α趋近于1时的点wise极限，证明其与梯度绝对值的Riesz势的关系，并扩展到更广的函数空间。
Bourgain et al.(2001) proved that for $p>1$ and smooth bounded domain $Ω\subseteq\mathbb{R}^N$, \begin{equation*} \lim\limits_{s\to1}(1-s)\iint \limits_{Ω\times Ω}\frac{\lvert f(x)-f(y) \rvert^p}{\lvert x-y \rvert^{N+sp}}dx dy=κ\int \limits_Ω\lvert \nabla f(x) \rvert^p dx \end{equation*} for all $f\in L^p(Ω)$. This gives a characterization of $W^{1,p}(Ω)$ by means of $W^{s,p}(Ω)$ seminorms only. For the case $p=1$, Dávila(2002) proved that when $Ω$ is a bounded domain with Lipschitz boundary, \begin{equation*} \lim\limits_{s\to1}(1-s)\iint \limits_{Ω\times Ω}\frac{\lvert f(x)-f(y) \rvert}{\lve…
Homogenization and averaging in the Bourgain-Brezis-Mironescu limit
该研究分析了周期系数二次分数阶能量在均匀化与Bourgain-Brezis-Mironescu极限同时作用下的Γ-极限，发现其由参数相对尺度决定，并在临界对数尺度下得到两种能量的凸组合。
A Bourgain-Brezis-Mironescu -type characterization for Sobolev differential forms
本文将 Bourgain-Brezis-Mironescu 关于 Sobolev 函数的 Besov 半范数极限刻画推广到微分形式，利用单纯形积分构造相应半范数，刻画弱外微分的 $L^p$ 正则性。
Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications
本文在任意开集上为 BV 函数建立 Bourgain--Brezis--Mironescu 公式，确定非局部能量 Γ-极限为全变差常数倍，并给出 BV 函数刻画及紧性结果。
A Bourgain-Brezis-Mironescu result for fractional thin films
研究分数薄层域上平方H^s-Gagliardo半范数的极限，证明在同时让ε→0和s→1时，乘以(1-s)ε^{2s-3}后仍收敛到几何缩减极限，探讨膜缩放与临界缩放关系。
We obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator. Furthermore we get a Bourgain-Brezis-Mironescu type limit formula for a related class of anisotropic nonlocal norms.
Extension Theorem and Bourgain--Brezis--Mironescu-Type Characterization of Ball Banach Sobolev Spaces on Domains
本文针对区域上的球 Banach Sobolev 空间建立了延拓定理，并利用外推法克服空间范数无显式表达等困难，证明了 Bourgain--Brezis--Mironescu 型渐近公式，进而实现了对该空间的刻画，且适用于多种 Sobolev 型空间。
On a weighted version of the BBM formula
本文证明了加权版 Bourgain-Brezis-Mironescu 公式在逐点与 Γ-收敛意义下成立，并给出能量有界序列的紧性准则，进而应用于 Poincaré 型不等式和第一特征值稳定性，还讨论了非局部类比。
Asymptotic Behaviour of fractional seminorms
该研究刻画了三类函数空间的渐近行为，得到了$s$趋近于1时的稳定性理论，回答了相关学者提出的问题，部分结果在$p=q$时为新成果。
Dimension reduction of fractional Sobolev seminorms on thin domains
研究薄域上Gagliardo半范数的渐进行为，揭示了不同阶数下的降维机制及临界指数s=1/2的局部极限特性。
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