Annalisa Cesaroni, Matteo Novaga
In this note we provide a second-order asymptotic expansion of the fractional perimeter Per$_s(E)$, as $s\to 1^-$, in terms of the local perimeter and of a higher order nonlocal functional.
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We review some recent results on minimisers of a non-local perimeter functional, in connection with some phase coexistence models whose diffusion term is given by the fractional Laplacian.
We introduce a notion of non-local perimeter which is defined through an arbitrary positive Borel measure on $\mathbb{R}^d$ which integrates the function $1\wedge |x|$. Such definition of non-local perimeter encompasses a wide range of perimeters which have been already studied in the literature, including fractional perimeters and anisotropic fractional perimeters. The main part of the article is devoted to the study of the asymptotic behaviour of non-local perimeters. As direct applications we recover well-known convergence results for fractional perimeters and anisotropic fractional perimet…
We prove a quantitative isoperimetric inequality for the Gaussian fractional perimeter using extension techniques. Though the exponent of the Fraenkel asymmetry is not sharp, the constant appearing in the inequality does not depend on the dimension but only on the Gaussian volume of the set and on the fractional parameter.
We consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume inte- gral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove existence and regularity of minimizers under suitable assumptions on the potential energy, which cover the periodic case. In the small volume regime we show that minimizers are close to balls, with a quantitative estimate.
This Master's thesis presents a study of the basic properties of the s-fractional perimeter and of the regularity theory of the corresponding s-minimal sets. In particular, we give full detailed proofs for all the Theorems contained in the article "Nonlocal Minimal Surfaces" by Caffarelli, Roquejoffre and Savin, where these s-minimal sets were introduced and studied for the first time.
Following \cite{Visintin}, we exploit the fractional perimeter of a set to give a definition of fractal dimension for its measure theoretic boundary. We calculate the fractal dimension of sets which can be defined in a recursive way and we give some examples of this kind of sets, explaining how to construct them starting from well known self-similar fractals. In particular, we show that in the case of the von Koch snowflake $S\subset\mathbb R^2$ this fractal dimension coincides with the Minkowski dimension, namely \begin{equation*} P_s(S)<\infty\qquad\Longleftrightarrow\qquad s\in\Big(0,2-\fra…
We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely $$ \int_\Om |\nabla u(x)|^2\,dx+\Per\Big(\{u > 0\},\Om \Big),$$ with $σ\in(0,1)$. We obtain regularity results for the minimizers and for their free boundaries $\p \{u>0\}$ using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems.
We consider a core-radius approach to nonlocal perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of $s$-fractional perimeter, defined for $0<s<1$, to the case $s\ge 1$\,. We show that, as the core-radius vanishes, such core-radius regularized $s$-fractional perimeters, suitably scaled, $Γ$-converge to the standard Euclidean perimeter. Under the same scaling, the first variation of such nonlocal perimeters gives back regularized $s$-fractional curvatures which, as the core radius vanishes, converge to the standard mean …
Gamma-convergence as $s\to1^-$ of anisotropic nonlocal fractional perimeter functionals
本文研究了各向异性非局部 $s$-分数阶周长泛函当 $s\to1^-$ 时的 $Γ(L^1_{loc})$-收敛渐近行为，证明了其重新缩放后收敛到合适的各向异性周长，且该结论在 $\mathbb{R}^n$ 和具有 Lipschitz 边界的有界区域上均成立。
We prove the $Γ$-convergence of the renormalised fractional Gaussian $s$-perimeter to the Gaussian perimeter as $s\to 1^-$. Our definition of fractional perimeter comes from that of the fractional powers of Ornstein-Uhlenbeck operator given via Bochner subordination formula. As a typical feature of the Gaussian setting, the constant appearing in front of the $Γ$-limit does not depend on the dimension.
Recently Frank and Seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from Sobolev seminorms of fractional order. This isoperimetric inequality is improved here in a quantitative form.
The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.
In the first part of this paper we show that a set $E$ has locally finite $s$-perimeter if and only if it can be approximated in an appropriate sense by smooth open sets. In the second part we prove some elementary properties of local and global $s$-minimal sets, such as existence and compactness. We also compare the two notions of minimizer (i.e. local and global), showing that in bounded open sets with Lipschitz boundary they coincide. However, in general this is not true in unbounded open sets, where a global $s$-minimal set may fail to exist (we provide an example in the case of a cylinder…
Sufficiently wide strips uniquely minimize the planar periodic fractional perimeter
该论文研究平面周期分数阶周长的极小化问题，证明足够宽的带状区域是唯一极小化者，并通过定量下界处理凹性困难，适用于任意非负可积剖面。
Stability of the ball in isoperimetric inequalities between two fractional perimeters
研究在s-周长和t-周长之间等周不等式中，球作为近球集的局部极小值问题，通过将F函数表示为参数函数u的泛函，并证明其关于Sobolev范数的稳定性结果。
This note treats several problems for the fractional perimeter or $s$-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as $s \nearrow 1$ is proven. It is shown that their limit as $s \searrow -\infty$ can be expressed in terms of the volume.
Sharp comparison between the perimeter and its fractional analogue in two dimensions
该研究证明在固定周长的平面集合中，任意阶分数阶周长以圆盘为最大，建立了尖锐比较结果。
We study the asymptotic behaviour of the renormalised $s$-fractional Gaussian perimeter of a set $E$ inside a domain $Ω$ as $s\to 0^+$. Contrary to the Euclidean case, as the Gaussian measure is finite, the shape of the set at infinity does not matter, but, surprisingly, the limit set function is never additive.
The anisotropic fractional $s$-perimeter with respect to a convex body $K$ in $\rn$ is shown to converge to the anisotropic perimeter with respect to the moment body of $K$ as $s\to 1^-$. A corresponding result is established for anisotropic fractional $s$-seminorms on $BV(\rn)$ (generalizing results of Bourgain, Brezis & Mironescu and Dávila). The minimizers of the anisotropic fractional $s$-isoperimetric inequality with respect to $K$ are shown to converge to the moment body of $K$. Anisotropic fractional Sobolev inequalities are established.
We consider here a nonlocal phase transition energy in a periodic medium and we construct solutions whose interfaces lie at a bounded distance from any given hyperplane. These solutions are either periodic or quasiperiodic, depending on the rational dependency of the normal direction to the reference hyperplane. Remarkably, the oscillations of the interfaces with respect to the reference hyperplane are bounded by a universal constant times the periodicity scale of the medium. This geometric property allows us to establish, in the limit, the existence of planelike nonlocal minimal surfaces in a…
We consider the minimization problem of the functional given by the sum of the fractional perimeter and a general Riesz potential, which is one generalization of Gamow's liquid drop model. We first show the existence of minimizers for any volumes if the kernel of the Riesz potential decays faster than that of the fractional perimeter. Secondly, we show the existence of generalized minimizers for any volumes if the kernel of the Riesz potential just vanishes at infinity. Finally, we study the asymptotic behavior of minimizers when the volume goes to infinity and we prove that a sequence of mini…
We study stable solutions to the fractional Allen-Cahn equation \linebreak $(-Δ)^{s/2} u = u-u^3$, $|u|<1$ in $\mathbb{R}^n$. For every $s\in (0,1)$ and dimension $n\geq 2$, we establish sharp energy estimates, density estimates, and the convergence of blow-downs to stable nonlocal $s$-minimal cones. As a consequence, we obtain a new classification result: if for some pair $(n,s)$, with $n\ge 3$, hyperplanes are the only stable nonlocal $s$-minimal cones in $\mathbb{R}^n\setminus\{0\}$, then every stable solution to the fractional Allen-Cahn equation in $\mathbb{R}^n$ is 1D, namely, its level …
We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set with smooth boundary, proving that they are {\em sufficiently close} to critical points of a suitable non-local potential. We then consider the fractional perimeter in half-spaces. We prove the existence of a minimizer under fixed volume constraint, showing some of its properties such as smoothness and symmetry, being a graph in the $x_N$-direction, and characterizing its intersection with the hyperplane $\{x_N=0\}$.
We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $Ω$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $Ω$. Moreover, we construct examples of sets for which the limit does not exist.
We show existence of fundamental domains which minimize a general perimeter functional in a homogeneous metric measure space. In some cases, which include the usual perimeter in the universal cover of a closed Riemannian manifold, and the fractional perimeter in $\mathbb R^n$, we can prove regularity of the minimal domains. As a byproduct of our analysis we obtain that a countable partition which is minimal for the fractional perimeter is locally finite and regular, extending a result previously known for the local perimeter. Finally, in the planar case we provide a detailed description of the…
We introduce a notion of fractional perimeter in an abstract Wiener space, and we show that halfspaces are the only volume-constrained minimisers.
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 …
Asymptotics as $s \to 0^+$ of the fractional perimeter on Riemannian manifolds
研究完备黎曼流形上分数阶拉普拉斯算子及分数阶周长在 $s\to 0^+$ 时的渐近行为，揭示其与有界调和函数存在性的联系，并推广至一般完备流形。
We prove existence of partitions of an open set $Ω$ with a given number of phases, which minimize the sum of the fractional perimeters of all the phases, with Dirichlet boundary conditions. In two dimensions we show that, if the fractional parameter $s$ is sufficiently close to $1$, the only singular minimal cone, that is, the only minimal partition invariant by dilations and with a singular point, is given by three half-lines meeting at $120$ degrees. In the case of a weighted sum of fractional perimeters, we show that there exists a unique minimal cone with three phases.
Regularity of sets of finite fractional perimeter and nonlocal minimal surfaces in metric measure spaces
本文在倍增度量测度空间中研究有限s-周长集合的正则性，给出周长有限性的充分条件并证明测度论边界的Hausdorff测度为零，同时推广非局部周长泛函并证明极小化子的存在性、一致密度性与多孔性。
This doctoral thesis is devoted to the analysis of some minimization problems that involve nonlocal functionals. We are mainly concerned with the $s$-fractional perimeter and its minimizers, the $s$-minimal sets. We investigate the behavior of sets having (locally) finite fractional perimeter and we establish existence and compactness results for (locally) $s$-minimal sets. We study the $s$-minimal sets in highly nonlocal regimes, that correspond to small values of the fractional parameter $s$. We introduce a functional framework for studying those $s$-minimal sets that can be globally written…
In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a cell motility model introduced in some previous work by the first author. We establish several facts about global minimizers with a volume constraint. In particular we prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either…
This note develops certain sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter.
We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $t$-perimeter and the $s$-perimeter, with $s$ smaller than $t$. Exploiting the quantitative fractional isoperimetric inequality, we show that balls are the unique minimizers if the volume is sufficiently small, depending on $t-s$, while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all $s,\,t$. When $s=0$ this problem reduces to the fractional isoper…
Given~$s,σ\in(0,1)$ and a bounded domain~$Ω\subset\R^n$, we consider the following minimization problem of $s$-Dirichlet plus $σ$-perimeter type $$ [u]_{ H^s(\R^{2n}\setminus(Ω^c)^2) } + \Per_σ\left(\{u>0\},Ω\right), $$ where~$[ \cdot]_{H^s}$ is the fractional Gagliardo seminorm and $\Per_σ$ is the fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones and a trivialization result for the flat case. Several classical free boundary problems are limit cases…
We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter converges exponentially fast to a single ball. As an intermediate result we establish a quantitative Alexandrov type estimate in the fractional setting for normal deformations of a ball. Finally, we provide existence for flat flows as limit points of the discrete flow when the time discretization parameter tends to zero.
A Boxing Inequality for the Fractional Perimeter
本文证明了分数阶周长的 Boxing 不等式，并据此推导出分数阶 Sobolev 空间中的迹不等式，统一包含 Sobolev 嵌入、Lorentz 空间改进及 Hardy 不等式，且在 $\alpha$ 趋于 $0$ 和 $1$ 时保持适当渐近性并恢复经典一阶不等式。
We consider a Dirichlet to Neumann operator $\mathcal{L}_a$ arising in a model for water waves, with a nonlocal parameter $a\in(-1,1)$. We deduce the expression of the operator in terms of the Fourier transform, highlighting a local behavior for small frequencies and a nonlocal behavior for large frequencies. We further investigate the $ Γ$-convergence of the energy associated to the equation $ \mathcal{L}_a(u)=W'(u) $, where $W$ is a double-well potential. When $a\in(-1,0]$ the energy $Γ$-converges to the classical perimeter, while for $a\in(0,1)$ the $Γ$-limit is a new nonlocal operator, tha…
Boxing inequalities for relative fractional perimeter and fractional Poincaré-type inequalities on John domains with the BBM factor
本文针对s-John域证明了带BBM因子的相对分数阶周长Boxing不等式及其泛函形式，推导了对应分数阶Poincaré型系列不等式，验证了John域条件的最优性，相关结果在Lipschitz域上也属全新。
$Γ$-convergence for higher order nonlocal phase transitions
该文研究高阶非局部相变能量泛函的$\Gamma$-收敛，证明当$0<s<3/4$时一阶变分贡献消失，极限为经典或分数阶周长，推翻曲率项预期。
Large-volume fractional isoperimetry on products of closed manifolds and Euclidean spaces
本文证明在闭流形与欧氏空间乘积上，当体积足够大时直圆柱唯一极小化分数周长，并给出周长亏损对偏差的控制，方法结合谱约化与切片重排。
On the Shape of Small Liquid Drops Minimizing Nonlocal Energies
本文研究势能作用下分数周长极小化问题中小液滴的平衡形状，并通过定量估计证明小体积极小元具有凸性且一致逼近球体。
Asymptotics and geometric flows for a class of nonlocal curvatures
本文研究了一类由对称径向核确定的非局部曲率在极限过程下的渐近行为，恢复了分数阶曲率等已知结果，并证明了相应水平集抛物型Cauchy问题粘性解的存在性、唯一性与稳定性。
In this note we obtain an asymptotic estimate for growth behavior of variational eigenvalues of the $p-$fractional eigenvalue problem on a smooth bounded domain with Dirichlet boundary condition.
We investigate the asymptotic behavior of positive ground states for Hénon type systems involving a fractional Laplacian on a bounded domain, when the powers of the nonlinearity approach the Sobolev critical exponent.
In this paper, as an improvement of the paper [K. Ishige, T. Kawakami and H. Michihisa, SIAM J. Math. Anal. 49 (2017) pp. 2167--2190], we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations.
In this paper, we prove the exact asymptotic behavior of singular positive solutions of fractional semi-linear equations $$(-Δ)^σu = u^p~~~~~~~~in ~~ B_1\backslash \{0\}$$ with an isolated singularity, where $σ\in (0, 1)$ and $\frac{n}{n-2σ} < p < \frac{n+2σ}{n-2σ}$.
A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies
针对有界域内带边界条件的分数阶非局部相变能量，研究证明其不存在二阶及阶数大于2-2s的分数阶Γ渐近展开，结果适用于任意空间维度与弱外部数据假设，为构建新的分数阶Γ收敛渐近展开提供了方向。
Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations.
In this paper we analyze the asymptotic behavior of the Dirichlet fractional Laplacian $(-Δ_{\mathbb R^{n+k}})^{s}$, with $s\in (0, 1)$, on bounded domains in $\mathbb R^{n+k}$ that become unbounded in the last $k$-directions. A dimension reduction phenomenon is observed and described via $Γ$-convergence.
We consider positive solutions of a fractional Lane-Emden type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we also prove that all the known uniqueness and nondegeneracy results in the local case extend to the nonlocal regime when the fractional parameter s is sufficiently close to 1.
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.
The self-similar asymptotics for solutions to the drift-diffusion equation with fractional dissipation, coupled to the Poisson equation, is analyzed in the whole space. It is shown that in the subcritical and supercritical cases, the solutions converge to the fractional heat kernel with algebraic rate. The proof is based on the entropy method and leads to a decay rate in the $L^1(\mathbb{R}^d)$ norm. The technique is applied to other semilinear equations with fractional dissipation.
In this paper, we analyze the asymptotic behavior of Palais-Smale sequences associated with fractional Yamabe type equations on an asymptotically hyperbolic Riemannian manifold. We prove that Palais-Smale sequences can be decomposed into the solution of the limit equation plus a finite number of bubbles, which are the rescaling of the fundamental solutions to the fractional Yamabe equation on Euclidean space. We also verify the non-interfering fact for multi-bubbles.
Asymptotic fractional uncertainty principle for the Helmholtz equation with periodic scattering data
本文研究了具有周期散射数据的Helmholtz方程解的分数阶色散，通过建立渐近分数阶不确定性原理，证明了在适当重新缩放下其不同频率间的相互作用与Schrödinger方程具有相同的波动行为。
This article discusses the analyticity and the long-time asymptotic behavior of solutions to space-time fractional diffusion equations in $\mathbb{R}^d$. By a Laplace transform argument, we prove that the decay rate of the solution as $t\to\infty$ is dominated by the order of the time-fractional derivative. We consider the decay rate also in a bounded domain.
This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $Ω\subset\mathbb{R}^n$, with or without a source term in the right hand side of the equation (commonly called chemical potential). Compare to the usual Allen-Cahn equation, the Laplace operator is here replaced by the fractional Laplacian $(-Δ)^s$ with $s\in(0,1/2)$, as defined in Fourier space. In the singular limit $\varepsilon\to 0$, we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surf…
We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to zero or to infinity.
In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity estimates of solution to the initial-boundary value problem are considered. Then combined with some important properties, including a maximum principle for a time-fractional ordinary equation and a coercivity inequality for fractional derivatives, the energy method shows that the decay in time of the solution is dominated by the term $t^{-α}$ as $t\to\infty$.
In this manuscript, we will study the asymptotic behavior for a class of nonlocal diffusion equations associated with the weighted fractional $\wp(\cdot)-$Laplacian operator involving constant/variable exponent. In the case of constant exponents, under some appropriate conditions, we will study the existence of solutions and asymptotic behavior of solutions by employing the subdifferential approach and we will study the problem when $\wp$ goes to $\infty$. Already, for case the weighted fractional $\wp(\cdot)$-Laplacian operator, we will also study the asymptotic behavior of the problem soluti…
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