A note on truncations in fractional Sobolev spaces

分数Sobolev对数不等式

Fractional Sobolev logarithmic inequalities

本文提出了一种分数Sobolev对数不等式，通过插值不等式和变分问题，扩展了经典Sobolev不等式到非局部框架，并提供了新的分析工具。

关于区域分数阶 Sobolev 不等式中的尖锐常数

On the sharp constants in the regional fractional Sobolev inequalities

该工作聚焦区域分数阶 Laplace 算子对应的分数阶 Sobolev 不等式，核心问题是刻画其中的尖锐常数。

Higher Sobolev regularity for the fractional $p-$Laplace equation in the superquadratic case

We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation $s$ reaches $1$.

带权重的分数Hardy-Sobolev和Hardy-Sobolev-Maz'ya不等式及其奇点在平坦子流形上的应用

Weighted fractional Hardy-Sobolev and Hardy-Sobolev-Maz'ya inequalities with singularities on flat submanifold

本文研究了带权重的分数Hardy-Sobolev和Hardy-Sobolev-Maz'ya不等式，探讨了奇点在平坦子流形上的应用，并证明了相关不等式在特定条件下的不成立性。

Fractional Hardy-Sobolev inequalities on half spaces

We investigate the existence of extremals for Hardy-Sobolev inequalities involving the Dirichlet fractional Laplacian of order s, 0<s<1, on half-spaces.

A density property for fractional weighted Sobolev spaces

In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.

Time fractional parabolic equations with measurable coefficients and embeddings for fractional parabolic Sobolev spaces

We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.

The effect of curvature in fractional Hardy--Sobolev inequality with the Spectral Dirichlet Laplacian

We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.

Fractional Sobolev inequalities associated with singular problems

In this paper we consider Sobolev inequalities associated with singular problems for the fractional $p$-Laplacian operator in a bounded domain of $\mathbb{R}^{N}$, $N\geq 2$.

On the Sobolev and Hardy constants for the fractional Navier Laplacian

We prove the coincidence of the Sobolev and Hardy constants relative to the "Dirichlet" and "Navier" fractional Laplacians of any real order $m\in(0,\frac{n}{2})$ over bounded domains in $\mathbb R^n$.

Basic results of fractional Orlicz-Sobolev space and applications to non-local problems

In this paper, we study the interplay between Orlicz-Sobolev spaces $L^{M}$ and $W^{1,M}$ and fractional Sobolev spaces $W^{s,p}$. More precisely, we give some qualitative properties of the new fractional Orlicz-Sobolev space $W^{s,M}$, where $s\in (0,1)$ and $M$ is an $N-$function. We also study a related non-local operator, which is a fractional version of the nonhomogeneous $M$-Laplace operator. As an application, we prove existence of weak solution for a non-local problem involving the new fractional $M-$Laplacian operator.

关于具有负可积幂的分数阶Sobolev函数零点集的注记

A note on zero sets of fractional sobolev functions with negative power of integrability

本文将Jiang和Lin提出的关于具有大零点集的函数的Poincaré型不等式推广到分数索伯列夫空间.作为结果，我们得到了逆可积的分数阶Sobolev函数零点集大小的豪斯多夫维数估计。对于次优的豪斯多夫维数估计，我们基于逐点Poincaré型不等式给出了一个完全初等的证明.

Limit of Fractional Power Sobolev Inequalities

We derive the Moser-Trudinger-Onofri inequalities on the 2-sphere and the 4-sphere as the limiting cases of the fractional power Sobolev inequalities on the same spaces, and justify our approach as the dimensional continuation argument initiated by Thomas P. Branson.

近场动力学与带变指数的各向异性分数阶Sobolev空间

Peridynamics and Anisotropic Fractional Sobolev Spaces with Variable Exponents

该数学分析领域研究构建近场动力学分数阶Sobolev空间及对应新型BBM型结果，探究带变指数的各向异性分数阶Sobolev空间的BBM型结果，还研究了各向同性情形的特征值问题。

Complete classification and nondegeneracy of minimizers for the fractional Hardy-Sobolev inequality, and applications

We study linear and non-linear equations related to the fractional Hardy--Sobolev inequality. We prove nondegeneracy of ground state solutions to the basic equation and investigate existence and qualitative properties, including symmetry of solutions to some perturbed equations.

Remarks on Sobolev norms of fractional orders

When a function belonging to a fractional-order Sobolev space is supported in a proper subset of the Lipschitz domain on which the Sobolev space is defined, how is its Sobolev norm as a function on the smaller set compared to its norm on the whole domain? On what do the comparison constants depend on? Do different norms behave differently? This article addresses these issues. We prove some inequalities and disprove some misconceptions by counter-examples.

分数 $p$-拉普拉斯 Hardy-Sobolev 运算符在有界域上的全局紧致性结果

Global compactness results for fractional $p$-Laplace Hardy Sobolev operator on a bounded domain

本文研究了分数 $p$-拉普拉斯 Hardy-Sobolev 运算符在有界域上的非线性临界问题，建立了全局紧致性结果。

径向函数在分数库仑-索博列夫空间中的加利亚尔多-尼伦伯格不等式

Gagliardo-Nirenberg Inequalities in Fractional Coulomb-Sobolev spaces for Radial functions

本文针对径向函数在分数库仑-索博列夫空间中扩展了加利亚尔多-尼伦伯格插值不等式的参数范围，并进一步分析了该扩展范围的最优性。

The Classical Obstacle problem with coefficients in fractional Sobolev spaces

We prove quasi-monotonicity formulae for classical obstacle-type problems with quadratic energies with coefficients in fractional Sobolev spaces, and a linear term with a Dini-type continuity property. These formulae are used to obtain the regularity of free-boundary points following the approaches by Caffarelli, Monneau and Weiss.

分数阶 Orlicz-Sobolev 空间的集中紧性原理

The concentration-compactness principle for fractional Orlicz-Sobolev spaces

该研究将集中紧性原理推广至分数阶 Orlicz-Sobolev 空间，并用以建立含分数阶 $g$-Laplacian 的临界椭圆问题弱解存在性。

The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis-Nirenberg problem

In this paper we extend the well-known concentration -- compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional $p-$laplacian in the whole ${\mathbb R}^n$.

Least-energy nodal solutions of nonlinear equations with fractional Orlicz-Sobolev spaces

In our work, we prove the existence of least-energy nodal solutions for nonlinear equations in which the new fractional Orlicz Laplacian is present. Precisely, we prove a compact embeddings result for weighted fractional Orlicz-Sobolev spaces. Next, by a minimization argument on Nehari manifold and a quantitative deformation lemma, we show our desired result.

H型群上精确的分数阶Sobolev及相关不等式

Sharp fractional Sobolev and related inequalities on H-type groups

本文确定了H型群上与次拉普拉斯算子共形不变分数阶幂相关的分数阶Sobolev不等式的精确常数，并由此导出精确的log-Sobolev不等式与Sobolev迹不等式，推广了前人结果。

分数阶索伯列夫空间的位似变体及其在Navier-Stokes系统中的应用

Homothetic variant of fractional Sobolev space with application to Navier-Stokes system revisited

本文对作者2007年DPDE论文\cite{Xiao}中的主要结果进行了更深入的理解。

关于区域中分数阶Orlicz-Sobolev半范数的Pólya-Szegö不等式的一个注记

A note on Pólya-Szegö inequality for fractional Orlicz-Sobolev seminorm in domains

本文证明了对称径向递减重排会增大区域中的分数阶Orlicz-Sobolev半范数，从而将Li-Wang的Pólya-Szegö型结果推广到比幂次更一般的Orlicz增长情形。

Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems

In the present paper, we deal with a new continuous and compact embedding theorems for the fractional Orlicz-Sobolev spaces, also, we study the existence of infinitely many nontrivial solutions for a class of non-local fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is $$(-\triangle)^{s}_{m}u+V(x)m(u)=f(x,u),\ x\in\mathbb{R}^{d},$$ where $0<s<1$, $d\geq2$ and $(-\triangle)^{s}_{m}$ is the fractional $M$-Laplace operator. The proof is based on the variant Fountain theorem established by Zou.

Introduction to fractional Orlicz-Sobolev spaces

In this paper, we define the fractional Orlicz-Sobolev spaces, and we prove some important results of these spaces. The main result is to show the continuous and compact embedding for these spaces. As an application, we prove the existence and uniqueness of a solution for a non local problem involving the fractional M-Laplacian operator.

Strauss and Lions type theorems for the fractional Sobolev spaces with variable exponent and applications to nonlocal Kirchhoff-Choquard problem

This paper deals with Strauss and Lions-type theorems for fractional Sobolev spaces with variable exponent $W^{s,p(.),\tilde{p}(.,.)}(Ω)$, when $p$ and $\tilde{p}$ satisfies some conditions. As application, we study the existence of solutions for a class of Kirchhoff-Choquard problem in $\mathbb{R}^N$.

Well-posedness of a porous medium flow with fractional pressure in Sobolev spaces

The nonnegative solution for a linear degenerate diffusion transport eqution is proved. As a result, we show the existence and uniqueness of the solution for the fractional porous medium equation in Sobolev spaces $H^α$ with nonnegative initial data, $α>\frac d2+1$. Besides, we correct a mistake in our previous paper \cite{zhou01}.

Compact embedding theorems and a Lions' type Lemma for fractional Orlicz-Sobolev spaces

In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions' "vanishing" Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of n…

分数阶各向异性Orlicz-Sobolev空间中的特征值问题

Eigenvalue problems in fractional anisotropic Orlicz-Sobolev spaces

本文引入分数阶各向异性Orlicz-Sobolev空间，利用变分方法研究含椭圆型非局部积分微分算子的分数阶各向异性问题，揭示了各向异性系数增长率竞争对特征值集合存在性与非存在性的关键影响。

非局部奇异椭圆问题在分数阶 Orlicz-Sobolev 空间中解的存在性与唯一性

Existence and uniqueness of solutions in fractional Orlicz-Sobolev spaces for a nonlocal singular elliptic problem

本文研究了一类带边界条件的奇异非线性椭圆问题，证明了其正解的存在性与唯一性，且该结果在分数阶 Orlicz-Sobolev 空间中成立。

Sobolev versus Hölder minimizers for the degenerate fractional $p$-Laplacian

We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Δ)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Ω$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(Ω)$ and in the weighted Hölder space $C^0_s(\overlineΩ)$, respectively, do coincide.

Fractional Hardy-Sobolev equations with nonhomogeneous terms

The paper deals with existence and multiplicity of positive solutions to nonlocal equations with critical Hrardy-Sobolev nonlinearities and external terms. We establish the profile decomposition of the Palais-Smale sequences associated with the functional and existence of at least two positive solutions to the equation.

The Brezis-Nirenberg problem for the fractional $p$-Laplacian involving critical Hardy-Sobolev exponents

We obtain existence, multiplicity, and bifurcation results for the Brezis-Nirenberg problem for the fractional $p$\nobreakdash-Laplacian operator, involving critical Hardy-Sobolev exponents. Our results are mainly extend results in the literature for $α=0$. In the absence of an explicit formula for a minimizer in the fractional Hardy-Sobolev inequality $α\not= 0$, we get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained in \cite{MM}.

关于分数阶各向异性Musielak-Sobolev空间及其在非局部特征值问题中的应用

On Fractional Anisotropic Musielak-Sobolev Spaces with Applications to Nonlocal Eigenvalue Problems

本文引入并研究分数阶各向异性Musielak-Sobolev空间，建立其基本性质与嵌入结果，将其应用于非局部各向异性特征值问题并证明特征值存在性，扩展并统一了相关现有模型。

A Bourgain-Brezis-Mironescu formula for anisotropic fractional Sobolev spaces and applications to anisotropic fractional differential equations

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.

时间分数阶演化方程的Sobolev空间与迹定理

Sobolev spaces and trace theorems for time-fractional evolution equations

该研究针对带Caputo分数阶导数的演化方程，在加权$L_p$空间框架下刻画适配的混合范数加权Sobolev与Besov空间，建立了迹定理与延拓定理，覆盖次扩散与超扩散情形，并给出解的初始行为的相关结论。

Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces

In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend the fractional Sobolev spaces $W^{s,p}$ to include the general case $W^sL_A$, where $A$ is an N-function and $s\in (0,1)$. We are concerned with some qualitative properties of the space $W^sL_A$ (completeness, reflexivity and separability). Moreover, we prove a continuous and compact embedding theorem of these spaces into Lebesgue spaces.

齐次分数阶 Sobolev 空间中的非局部临界椭圆方程

Nonlocal critical elliptic equations in homogeneous fractional Sobolev spaces

本文基于 ${\mathbb Z}_2$-上同调指标的抽象临界点定理、分数阶 $p$-Laplacian 方程的正则性结果与紧嵌入，证明了齐次分数阶 Sobolev 空间中非局部临界增长椭圆方程的多重性结果。

On density of smooth functions in weighted fractional Sobolev spaces

We prove that smooth $C^\infty$ functions are dense in weighted fractional Sobolev spaces on an arbitrary open set, under some mild conditions on the weight. We also obtain a~similar result in non-weighted spaces defined by some kernel similar to $x\mapsto |x|^{-d-sp}$. One may consider the results to be a~version of the Meyers--Serrin theorem.

A Global Compactness type result for Palais-Smale sequences in fractional Sobolev spaces

We extend the Global Compactness result by M. Struwe (Math. Z, 1984) to any fractional Sobolev spaces $\dot{H}^s(Ω)$ for $0<s<N/2$ and $Ω\subset \mathbb{R}^N$ a bounded domain with smooth boundary. The proof is a simple direct consequence of the so-called Profile Decomposition of P. Gerard (ESAIM: Control, Optimisation and Calculus of Variations, 1998).

The distributional hyper-Jacobian determinants in fractional Sobolev spaces

In this paper we give a positive answer to a question raised by Baer-Jerison in connection with hyper-Jacobian determinants and associated minors in fractional Sobolev spaces. Inspired by recent works of Brezis-Nguyen and Baer-Jerison on the Jacobian and Hessian determinants, we show that the distributional $m$th-Jacobian minors of degree $r$ are weak continuous in fractional Sobolev spaces $W^{m-\frac{m}{r},r}$, and the result is optimal, satisfying the necessary conditions, in the frame work of fractional Sobolev spaces. In particular, the conditions can be removed in case $m=1,2$, i.e., the…

On density of compactly supported smooth functions in fractional Sobolev spaces

We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $W^{s,p}(Ω)$ for an open, bounded set $Ω\subset\mathbb{R}^{d}$. The density property is closely related to the lower and upper Assouad codimension of the boundary of $Ω$. We also describe explicitly the closure of $C_{c}^{\infty}(Ω)$ in $W^{s,p}(Ω)$ under some mild assumptions about the geometry of $Ω$. Finally, we prove a variant of a fractional order Hardy inequality.

On Fractional Schrodinger Equations in sobolev spaces

Let $σ\in(0,1)$ with $σ\neq\frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in $\mathbb R^d$: $$i\partial_tu+(-Δ)^σu+μ|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-Δ)^σ$ is the Fourier multiplier of symbol $|ξ|^{2σ}$, and $μ=\pm 1$. This model has been introduced by Laskin in quantum physics \cite{laskin}. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.

Infinitely many solutions for a class of fractional Orlicz-Sobolev Schrödinger equations

In the present paper, we deal with a new compact embedding theorem for a subspace of the new fractional Orlicz-Sobolev spaces. We also establish some useful inequalities which yields to apply the variational methods. Using these abstract results, we study the existence of infinitely many nontrivial solutions for a class of fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is $$(-\triangle)^{s}_{m}+V(x)m(u)u=f(x,u),\ x\in\mathbb{R}^{N},$$ where $s\in ]0,1[$, $N\geq2$, $(-\triangle)^{s}_{m}$ is fractional $M$-Laplace operator and the nonlinearity $f$ is sublinear as $|u| \…

有界光滑区域中分数阶Sobolev空间内Euler方程的可解性

Solvability of Euler equations in the fractional Sobolev spaces in a bounded smooth domain

本文利用能量方法和特征线法结合椭圆估计，证明了有界光滑区域上分数阶Sobolev空间中Euler方程的适定性，其中二维情形时间全局、三维情形时间局部。

A Pólya-Szegö principle for general fractional Orlicz-Sobolev spaces

In this article we prove modular and norm Pólya-Szegö inequalities in general fractional Orlicz-Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of theses spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary we prove a Rayleigh-Faber-Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators.

分数阶Sobolev空间中具有光滑子流形奇点的Trudinger-Moser型不等式

Trudinger-Moser type inequality in fractional Sobolev space with singularity on smooth submanifold

研究分数阶Sobolev空间中在光滑子流形上具有奇点的Trudinger-Moser型不等式，通过分数阶Hardy不等式证明并展示常数尖锐性，同时建立奇点集上消失的分数阶Sobolev空间等价性。

Fractional Hardy-Sobolev elliptic problems

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-Δ)^{\frac α2} u=λ|u|^{r-2}u+μ\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in }\quad Ω, \\ \\ u=0 &{\rm on }\quad \partialΩ, \end{array} \right. \end{equation} where $Ω$ is a smooth bounded domain in $\mathbb R^N$ with $0\in Ω$, $λ,μ>0,0<s\leqα$, $(-Δ)^{\frac α2}$ is the fractional Laplacian operator with $0<α<2$. We establish existence results of problem \eqref{eq:1} for subcritical, Sobolev critical and Hardy-Sobolev critical cases.

On the fractional Musielak-Sobolev spaces in R^d: Embedding results & applications

This paper deals with new continuous and compact embedding theorems for the fractional Musielak-Sobolev spaces in $\mathbb{R}^d$. As an application, using the variational methods, we obtain the existence of nontrivial weak solution for the following Schrödinger equation $$ (-Δ)_{g_{x,y}}^s u+V(x)g(x,x,u)=b(x)\vert u\vert^{p(x)-2}u,\ \text{for all}\ x\in \mathbb{R}^d,$$ where $(-Δ)_{g_{x,y}}^s$ is the fractional Museilak $g_{x,y}$-Laplacian, $V$ is a potential function, $b\in L^{δ^{'}(x)}(\mathbb{R}^d)$, and $p,δ\in C\left(\mathbb{R}^d,(1,+\infty)\right)\cap L^{\infty}(\mathbb{R}^d)$. We would …

局部有限图上的分数阶 Sobolev 空间与分数阶 $p$-Laplace 方程

Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs

针对一般图上分数阶问题研究不足，本文在连通、局部有限且随机完备图上构造分数阶 Sobolev 空间并提出分数阶 $p$-Laplace 算子，证明空间性质并研究相关非线性 Schrödinger 型方程。

Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces

In the first part of this article we deal with the existence of at least three non-trivial weak solutions of a nonlocal problem with nonstandard growth involving a nonlocal Robin type boundary condition. The second part of the article is devoted to study eigenvalues and minimizers of several nonlocal problems for the fractional $g-$Laplacian $(-Δ_g)^s$ with different boundary conditions, namely, Dirichlet, Neumann and Robin.

带分数阶Sobolev范数的非线性插值不等式与生物膜中的斑图形成

Nonlinear interpolation inequalities with fractional Sobolev norms and pattern formation in biomembranes

本文研究生物膜斑图形成的一维变分模型，推导最小能量标度律，并提出用Cahn-Hillard/Modica-Mortola能量界定分数阶Sobolev半范数的新非线性插值不等式作为主要工具。