Federico Buseghin, Nicola Garofalo, Giulio Tralli
In this note we study the behaviour as $s\to 0^+$ of some semigroup based Besov seminorms associated with a non-symmetric and hypoelliptic diffusion with a drift. Our results generalise a previous one of Maz'ya and Shaposhnikova for the classical fractional Sobolev spaces $W^{s,p}$, and they also underscore a new phenomenon caused by the presence of the drift.
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The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space. In this article, we relate these two types of variational problems. In particular, we investigate the connection between the nonlocal minimal surfaces and the minimisers of the $W^{s,1}$-seminorm. In particular, we show that a function is a minimiser for the fract…
On Limit Formulas for Besov Seminorms and Nonlocal Perimeters in the Dunkl Setting
本文在 Dunkl 框架下推广了 Besov 半范数极限公式与非局部周长极限公式，提出了无维数依赖的 Maz'ya--Shaposhnikova 公式，并弱化了边界正则性条件，结果在经典拉普拉斯情形下亦为新发现。
Second-order asymptotics of fractional Gagliardo seminorms as $s\to1^-$ and convergence of the associated gradient flows
本文研究分数阶 Gagliardo 半范数在 $s\ o1^-$ 时的二阶渐近展开，证明能量泛函的 Mosco 收敛性，并由此得到相应 $L^2$-梯度流的收敛性。
We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function $u$ on $\mathbb R^n$ and of its perturbation $uφ_μ$, where $φ_μ$ is a suitably chosen eigenfunction of the Laplace-Beltrami operator on the sphere $\mathbb S^{n-1}$, thus providing a technical but rather powerful tool to detect symmetry breaking and multiplicity phenomena in variational equations driven by the fractional Laplace operator. A concrete application to a problem related to the fractional Caffarelli-Kohn-Nirenberg inequality is given.
We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and $s$ is close to 1.
Age-structured models with nonlocal diffusion arise naturally in describing the population dynamics of biological species and the transmission dynamics of infectious diseases in which individuals disperse nonlocally and interact each other and the age structure of individuals matters. In the first part of our series papers, we study the principal spectral theory of age-structured models with nonlocal diffusion of Dirichlet type. First, we provide two criteria on the existence of principal eigenvalues by using the theory of resolvent positive operators with their perturbations. Then we define t…
Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case
本文研究二维分数阶 Sobolev 极限情形下 Schrödinger-Poisson 系统的解的存在性与定性性质，通过辅助 Choquard 方程的变分逼近方法证明解存在，并利用移动平面技巧得到对称性与衰减结果。
The principal eigenvalue for linear elliptic operator has been known to be one of very useful tools to investigate many important partial differential equations. Due to the pioneering works of Berestycki et al. \cite{BCV1,BCV2}, the study of qualitative properties for the principal eigenvalue of nonlocal operators has attracted a lot of attention of the community from theory to application (For examples \cite{LL22-1,LLS22,LZ1,LZ2,SLLY,DDF,XLR}). In this paper, motivated from the study of mathematical modeling the dynamics of infectious diseases in \cite{NV1, ZZLD, WD}, we analyze the asymptoti…
We establish the $Γ$-convergence of some energy functionals describing nonlocal attractive interactions in bounded domains. The interaction potential solves an elliptic equation (local or nonlocal) in the bounded domain and the primary interest of our results is to identify the effects that the boundary conditions imposed on the potential have on the limiting functional. We consider general Robin boundary conditions, which include Dirichlet and Neumann conditions as particular cases. Depending on the order of the elliptic operator the limiting functional involves the usual perimeter or some fr…
Nonlocal Schrödinger-Poisson systems in $\mathbb R^N$: the fractional Sobolev limiting case
本文研究全空间中非局部 Schrödinger-Poisson 系统在分数阶 Sobolev 极限情形下的正解存在性，通过变分逼近方法处理符号变化对数 Riesz 核带来的困难，并建立了解的对称性、正则性与衰减性。
Non-local evolution equations with Lévy diffusion: Well-posedness and limiting behavior
研究具有 Lévy 扩散的时间非局部演化方程，通过从属原理和松弛函数理论，在相关条件下探讨非局部线性演化问题经典解存在性、向广义 Rayleigh - Stokes 方程的极限情况及非线性问题温和解的性质。
On De Giorgi's Conjecture of Nonlocal approximations for free-discontinuity problems: The symmetric gradient case
本研究将De Giorgi关于自由不连续问题非局部逼近的猜想推广至对称梯度泛函情形，通过推广Fréchet-Kolmogorov方法证明能量等有界形变的紧性与Gamma收敛，借助GSBD空间新刻画确定极限空间，建立极限表示。
The Kuznetsov and Blackstock equations of nonlinear acoustics with nonlocal-in-time dissipation
本文研究含分数阶时间非局部耗散的 Kuznetsov 与 Blackstock 方程，给出记忆核最小假设，并证明关于小参数一致的适定性及能量范数收敛速率。
On parameterized nonlocal-fractional transmission problems and associated function spaces
针对跨界面耦合的非局部与分数阶模型，论文由参数化能量变分建立严格数学框架，证明适定性并分析多种极限行为。
Nonlocal basis pursuit: Nonlocal optimal design of conductive domains in the vanishing material limit
本文研究导电材料量趋于零时非局部最优设计问题的极限，证明该极限问题为凸的基追踪型问题，在混合Lebesgue空间中可解，并通过放弃两点通量反对称性要求建立了与局部测度值问题的极限关系。
Principal eigenvalues of nonlocal operators with advection: sharp scaling limits and spectral phase transitions
该研究针对带固定符号漂移的一维非局部扩散算子，通过直接方法和多种分析技术确定其主特征值的精确标度极限与谱相变，填补了非自伴情形下变分刻画缺失的研究空白。
This paper is concerned with qualitative properties of solutions to nonlocal reaction-diffusion equations of the form$$ \int\_{\mathbb{R}^N\setminus K} J(x-y)\,\big( u(y)-u(x) \big)\,\D y+f(u(x))=0, \quad x\in\R^N\setminus K,$$set in a perforated open set $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a bounded compact "obstacle" and $f$ is a bistable nonlinearity. When $K$ is convex, we prove some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on $K$.
In this paper, we proceed to study the nonlocal diffusion problem proposed by Li and Wang [8], where the left boundary is fixed, while the right boundary is a nonlocal free boundary. We first give some accurate estimates on the longtime behavior by constructing lower solutions, and then investigate the limiting profiles of this problem when the expanding coefficient of free boundary converges to $0$ and $\yy$, respectively. At last, we focus on two important kinds of kernel functions, one of which is compactly supported and the other behaves like $|x|^{-γ}$ with $γ\in(1,2]$ near infinity. With…
We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the paper, we show that these functionals are indeed perimeters in a generalised sense and we establish existence of minimisers for the corresponding Plateau problem. Also, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe flat boundary conditions. A Gamma-convergence result is discussed in the second …
Given an open and bounded set $Ω\subset\mathbb{R}^N$, we consider the problem of minimizing the ratio between the $s-$perimeter and the $N-$dimensional Lebesgue measure among subsets of $Ω$. This is the nonlocal version of the well-known Cheeger problem. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and nonlocal eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.
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。
Time Asymptotics and Scaling Limits for a Nonlocal Fokker-Planck Equation with Heavy-Tailed Kernel
本文研究了非局部福克-普朗克方程在重尾核下的时间渐近性与标度极限，证明了解向平衡状态的指数收敛性，并展示了在参数趋于零或一的情况下的一致收敛性。
We connect boundary conditions for one-sided pseudo-differential operators with the generators of modified one-sided Lévy processes. On one hand this allows modellers to use appropriate boundary conditions with confidence when restricting the modelling domain. On the other hand it allows for numerical techniques based on differential equation solvers to obtain fast approximations of densities or other statistical properties of restricted one-sided Lévy processes encountered, for example, in finance. In particular we identify a new nonlocal mass conserving boundary condition by showing it corre…
We connect boundary conditions for one-sided pseudo-differential operators with the generators of modified one-sided Lévy processes. On one hand this allows modellers to use appropriate boundary conditions with confidence when restricting the modelling domain. On the other hand it allows for numerical techniques based on differential equation solvers to obtain fast approximations of densities or other statistical properties of restricted one-sided Lévy processes encountered, for example, in finance. In particular we identify a new nonlocal mass conserving boundary condition by showing it corre…
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