Pekka Koskela, Alexander Ukhlov, Zheng Zhu
Let $n\geq 2$ and $1\leq q<p<\fz$. We prove that if $Ω\subset\mathbb R^n$ is a Sobolev $(p, q)$-extension domain, with additional capacitory restrictions on boundary in the case $q\leq n-1$, $n>2$, then $|\partialΩ|=0$. In the case $1\leq q<n-1$, we give an example of a Sobolev $(p,q)$-extension domain with $|\partialΩ|>0$.
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In this note, we prove that the boundary of a $(W^{1, p}, BV)$-extension domain is of volume zero under the assumption that the domain $\boz$ is $1$-fat at almost every $x\in\partial\boz$. Especially, the boundary of any planar $(W^{1, p}, BV)$-extension domain is of volume zero.
Zero-extension convergence and Sobolev spaces on changing domains
本文在不假设参考构型的前提下，将弱收敛与强收敛推广到底层区域同时变化的Sobolev函数序列，建立相应理论与紧性定理，并通过实例验证其有效性。
Extensions of traces for Sobolev mappings into manifolds at the endpoint $p=1$
该研究针对边界紧的黎曼流形到完备黎曼流形的$W^{1,1}$型Sobolev映射，通过二进立方体格铺砌定义域的方法构造延拓，直接证明了迹与延拓定理，还得到了映射$L^1$能量的渐近刻画，结果可推广至半空间与带状区域。
We study in this article some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended without the use of the Littlewood-Paley decomposition to this general framework.
The extension of traces for Sobolev mappings between manifolds
本文刻画了使一阶Sobolev映射空间到分数阶Sobolev-Slobodecki\uı空间的迹算子为满射的紧致黎曼流形条件，并利用新构造在无已知障碍时实现迹的延拓，同时给出线性估计。
This paper extends a stability estimate of the Sobolev Inequality established by G. Bianchi and H. Egnell in their paper "A note on the Sobolev Inequality." Bianchi and Egnell's Stability Estimate answers the question raised by H. Brezis and E. H. Lieb: "Is there a natural way to bound $\| \nabla φ\|_2^2 - C_N^2 \| φ\|_\frac{2N}{N-2}^2$ from below in terms of the 'distance' of $φ$ from the manifold of optimizers in the Sobolev Inequality?" Establishing stability estimates - also known as quantitative versions of sharp inequalities - of other forms of the Sobolev Inequality, as well as other sh…
Given $n \in \mathbb{N}_*$, a compact Riemannian manifold $M$ and a Sobolev map $u \in W^{n/(n + 1), n + 1} (\mathbb{S}^n; M)$, we construct a map $U$ in the Sobolev-Marcinkiewicz (or Lorentz-Sobolev) space $W^{1, (n + 1, \infty)} (\mathbb{B}^{n + 1}; M)$ such that $u = U$ in the sense of traces on $\mathbb{S}^{n} = \partial \mathbb{B}^{n + 1}$ and whose derivative is controlled: for every $λ> 0$, $$ λ^{n + 1} \big\vert\big\{ x \in \mathbb{B}^{n + 1} : \vert D U (x)\vert > λ\big\}\big\vert \le γ\Big(\int_{\mathbb{S}^n}\int_{\mathbb{S}^n} \frac{\vert u (y) - u (z)\vert^{n + 1}}{\vert y - z\vert…
On the local character of the extension of traces for Sobolev mappings
本文证明Sobolev映射的迹延拓具有局部性质，即延拓存在的障碍在全局情形下纯粹是局部的。
In this paper we apply estimates of the norms of Sobolev extension operators to the spectral estimates of of the first nontrivial Neumann eigenvalue of the Laplace operator in non-convex extension domains. As a consequence we obtain a connection between resonant frequencies of free membranes and the smallest-circle problem (initially proposed by J.~J.~Sylvester in 1857).
Volume growths versus Sobolev inequalities
本文研究满足Sobolev型不等式的度量测度空间的体积增长，证明了相关定量体积增长结果并部分解答Ledoux的问题，还在CD(0,N)空间中得到精确插值不等式及等号刻画。
Sobolev $(p,q)$-extension operators and Neumann eigenvalues
本文研究了Sobolev空间上的$(p,q)$-扩展算子，通过构造向外尖突域的扩展算子并估计其范数，证明了$p$-Laplace算子在向外尖突域中的非线性Neumann特征值的估计。
We prove the existence of Sobolev extension operators for certain uniform classes of domains in a Riemannian manifold with an explicit uniform bound on the norm depending only on the geometry near their boundaries. We use this quantitative estimate to obtain uniform Neumann heat kernel upper bounds and gradient estimates for positive solutions of the Neumann heat equation assuming integral Ricci curvature conditions and geometric conditions on the boundary. Those estimates also imply quantitative lower bounds on the first Neumann eigenvalue of the considered domains.
On the existence of extensions for manifold-valued Sobolev maps on perforated domains
针对穿孔域上的流形值Sobolev函数，结合经典扩展结果与基于流形拓扑的收缩论证构造扩展，并探讨其与迹算子满射性的关联，为扩展存在性提供必要条件。
Singular extension of critical Sobolev mappings under an exponential weak-type estimate
本文针对临界Sobolev空间中的球面到紧黎曼流形映射，构造了满足指数弱型Sobolev估计的球内部延拓，并将结果推广到半空间和双曲空间情形。
Gaussian upper bounds, volume doubling and Sobolev inequalities on graphs
本文研究图上Sobolev不等式与高斯热核上界及体积倍增条件的等价性，通过引入局部正则性条件和可变修正函数，建立了Sobolev维数与倍增维数及顶点度增长之间的联系。
Using the quasiconformal mappings theory and Sobolev extension operators, we obtain estimates of principal frequencies of free non-homogeneous membranes. The suggested approach is based on connections between divergence form elliptic operators and quasiconformal mappings. Free non-homogeneous membranes we consider as circular membranes in the quasiconformal geometry associated with non-homogeneity of membranes. As a consequence we get a connection between principal frequencies of free membranes and the smallest-circle problem (initially suggested by J.~J. Sylvester in 1857).
We give a quantitative characterization of traces on the boundary of Sobolev maps in $\dot{W}^{1,p}(\mathcal M, \mathcal N)$, where $\mathcal{M}$ and $\mathcal{N}$ are compact Riemannian manifolds, $\partial \mathcal{M} \neq \emptyset$: the Borel-measurable maps $u\colon \partial \mathcal M \to \mathcal{N}$ that are the trace of a map $U\in \dot{W}^{1,p}(\mathcal M, \mathcal{N})$ are characterized as the maps for which there exists an extension energy density $w \colon \partial \mathcal{M} \to [0,\infty]$ that controls the Sobolev energy of extensions from $\lfloor p - 1 \rfloor$-dimensional s…
We give a necessary condition for a domain to have a bounded extension operator from $L^{1,p}(Ω)$ to $L^{1,p}(\mathbb R^n)$ for the range $1 < p < 2$. The condition is given in terms of a power of the distance to the boundary of $Ω$ integrated along the measure theoretic boundary of a set of locally finite perimeter and its extension. This generalizes a characterizing curve condition for planar simply connected domains, and a condition for $W^{1,1}$-extensions. We use the necessary condition to give a quantitative version of the curve condition. We also construct an example of an extension dom…
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…
The author establishes some geometric criteria for a Haj\lasz-Sobolev $\dot M^{s,\,p}_\ball$-extension (resp. $\dot M^{s,\,p}_\ball$-imbedding) domain of ${\mathbb R}^n$ with $n\ge2$, $s\in(0,\,1]$ and $p\in[n/s,\,\infty]$ (resp. $p\in(n/s,\,\infty]$). In particular, the author proves that a bounded finitely connected planar domain $\boz$ is a weak $α$-cigar domain with $α\in(0,\,1)$ if and only if $\dot F^s_{p,\,\infty}({\mathbb R}^2)|_\boz=\dot M^{s,\,p}_\ball(\boz)$ for some/all $s\in[α,\,1)$ and $p=(2-\az)/(s-α)$, where $\dot F^s_{p,\,\infty}({\mathbb R}^2)|_\boz$ denotes the restriction o…
We show that every $\mathbb{R}^d$-valued Sobolev path with regularity $α$ and integrability $p$ can be lifted to a Sobolev rough path provided $α< 1/p<1/3$. The novelty of our approach is its use of ideas underlying Hairer's reconstruction theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.
The Hessian Sobolev inequality of X.-J. Wang, and the Hessian Poincaré inequalities of Trudinger and Wang are fundamental to differential and conformal geometry, and geometric PDE. These remarkable inequalities were originally established via gradient flow methods. In this paper, direct elliptic proofs are given, and extensions to trace inequalities with general measures in place of Lebesgue measure are obtained. The new techniques rely on global estimates of solutions to Hessian equations in terms of Wolff's potentials, and duality arguments making use of a non-commutative inner product on th…
Classification of positive solutions of Hardy-Sobolev equation without the finite volume constraints
本文通过构造向量场积分不等式和牛顿型不等式，对无有限体积约束的Hardy-Sobolev方程正解进行了分类，扩展了Ou的结果。
In this article, we develop the theory of weighted $L^2$ Sobolev spaces on unbounded domains in $\mathbb R^n$. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the bounded, unweighted case.
Geometric rigidity on Sobolev spaces with variable exponent and applications
该研究将刚性估计与Korn不等式推广到变指数Sobolev空间框架，借助覆盖论证、指数log-Hölder连续性及类Nitsche延拓完成证明，并通过Γ-收敛实现次二次增长下非线性到线弹性的过渡。
In this paper we fix $1\le p<\infty$ and consider $(\Om,d,μ)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $μ$ supporting a $p$-Poincaré inequality such that $\Om$ is a uniform domain in its completion $\bar\Om$. We realize the trace of functions in the Dirichlet-Sobolev space $D^{1,p}(\Om)$ on the boundary $\partial\Om$ as functions in the homogeneous Besov space $HB^α_{p,p}(\partial\Om)$ for suitable $α$; here, $\partial\Om$ is equipped with a non-atomic Borel regular measure $ν$. We show that if $ν$ satisfies a $θ$-codimensional condit…
A Survey on the Div-Curl Lemma and Some Extensions to Fractional Sobolev Spaces
本文是一本待出版著作的章节，回顾Div-Curl引理的经典表述与证明，在Riesz分数阶梯度与散度框架下介绍该引理在分数阶Sobolev空间中的若干推广。
We consider the minimizers of the energy $$ \|u\|_{H^s(Ω)}^2+\int_ΩW(u)\,dx,$$ with $s \in (0,1/2)$, where $\|u\|_{H^s(Ω)}$ denotes the total contribution from $Ω$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer $u$ in a large ball $B_R$ occupy a volume comparable with the volume of $B_R$.
Sobolev and Michael-Simon inequalities via the ABP method beyond Euclidean volume growth
本文通过改进Brendle的接触集论证，在体积非坍塌假设下发展了ABP方法，证明了包含低阶项的Michael-Simon型不等式和Varopoulos的L1-Sobolev不等式，并给出了等周轮廓的下界。
In this paper, we are concerned with some qualitative properties of the new fractional Musielak-Sobolev spaces $W^sL_{\varPhi_{x,y}}$ such that the generalized Poincaré type inequality and some continuous and compact embedding theorems of these spaces. Moreover, we prove that any function in $W^sL_{\varPhi_{x,y}}(Ω)$ may be extended to a function in $W^sL_{\varPhi_{x,y}}(\R^N)$, with $Ω\subset \R^N$ is a bounded domain of class $C^{0,1}$. In addition, we establish a result relates to the complemented subspace in $W^s{L_{\varPhi_{x,y}}}\left( \R^N\right)$. As an application, using the mountain …
Failure of zero extension in parabolic Sobolev spaces
构造反例证明抛物Sobolev空间中空间零延拓可能失败，缺陷源于初始-边界角点处的自相似边界层，并讨论散度型抛物方程解空间的适用性。
Sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature
研究非紧致黎曼流形上带有有界瑞奇曲率的尖锐Sobolev不等式，通过等周轮廓的渐进行展和局部Sobolev不等式推导
Singular extension of critical Sobolev mappings with values into complete Riemannian manifolds
基于Petrunin的判据，将临界Sobolev映射的奇异延拓结果从欧氏空间推广到具有正触及完备非紧黎曼流形，并证明延拓映射满足指数弱型Sobolev-Marcinkiewicz估计。
This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.
There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $λ=n-α$ (that is for the case of $α>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via…
The purpose of this paper is to characterize all the entire solutions of the homogeneous Helmholtz equation (solutions in $\mathbb{R}^d$) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^α(\mathbb{S}^{d-1}),$ with $α\in \mathbb{R}$. We present two characterizations. The first one is written in terms of certain $L^2$-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For $α>0$ this characterization involves a …
In this paper, we investigate cylindrical extensions of critical Sobolev type (improved Hardy) inequalities and identities in the style of Badiale-Tarantello [BT02], which in a special case give a critical Hardy inequality and its stability results. We also obtain higher-order identities, which interestingly include well-known numbers like double factorial, Oblong numbers, and Stirling numbers of the second kind. All functional identities are obtained in $L^{p}$ for $p\in (1,\infty)$ without the real-valued function assumption, which gives a simple and direct understanding of the corresponding…
In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domains.
We review the current state of the art concerning the characterization of traces of the spaces $W^{1, p} (\mathbb{B}^{m-1}\times (0,1), \mathcal{N})$ of Sobolev mappings with values into a compact manifold $\mathcal{N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p < m$ is an integer and the homotopy group $π_p (\mathcal{N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $π_1 (\mathcal{N})$ is finite and $π_2 (\mathcal{N}) \simeq \dotsb \simeq π_{\lfloor p - 1 \rfloor} (\mathcal{…
Two-scale density of almost smooth functions in sphere-valued Sobolev spaces: A high-contrast extension of the Bethuel-Zheng theory
本文将 Bethuel-Zheng 的开创性论证推广至双尺度设定，证明了球面值 Sobolev 空间中映射的强双尺度逼近结果，并应用于高对比度微磁学的变分问题。
Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier-Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) en…
Sharp Sobolev inequalities on noncompact Riemannian manifolds with ${\sf Ric}\geq 0$ via Optimal Transport theory
本文基于 Optimal Mass Transportation 理论，在具有非负 Ricci 曲率的非紧黎曼流形上建立了 sharp $L^p$-Sobolev 和对数 Sobolev 不等式，并为体积非塌陷估计提供了初等证明。
Let $X$ be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincaré inequality. We study extensions of Newtonian Sobolev functions to the completion $\widehat{X}$ of $X$ and use them to obtain several results on $X$ itself, in particular concerning minimal weak upper gradients, Lebesgue points, quasicontinuity, regularity properties of the capacity and better Poincaré inequalities. We also provide a discussion about possible applications of the completions and extension results to $p$-harmonic functions on noncomplete spaces and show by examples…
The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.
Let $Ω\subset\dR^d$ be a bounded or an unbounded Lipschitz domain. In this note we address the problem of continuation of functions from the Sobolev space $H^1(Ω)$ up to functions in the Sobolev space $H^1(\dR^d)$ via a linear operator. The minimal possible norm of such an operator is estimated from below in terms of spectral properties of self-adjoint Robin Laplacians on domains $Ω$ and $\dR^d\setminus\ovΩ$. Another estimate of this norm is also given, where spectral properties of Schrödinger operators with the $δ$-interaction supported on the hypersurface $\partialΩ$ are involved. General re…
Classification of positive solutions of critical anisotropic Sobolev equation without the finite volume constraint
本文通过新方法和Kato不等式，无有限体积约束下对临界各向异性Sobolev方程正解进行分类，并揭示了临界指数$p$对$n$和$\Lambda$的依赖关系。
Sobolev--Morrey Spaces and Divergence-Form Degenerate Second-Order Elliptic Equations on Domains with Higher Co-Dimensional Boundaries
研究\(\mathbb{R}^n\)中高余维边界区域上加权齐次Sobolev - Morrey空间，建立实变理论，应用迹和延拓算子有界性，获散度型退化二阶椭圆方程Dirichlet问题加权先验估计。
Let $u(\cdot,\cdot)$ be the Caffarelli-Silvestre extension of $f.$ The first goal of this article is to establish the fractional trace type inequalities involving the Caffarelli-Silvestre extension $u(\cdot,\cdot)$ of $f.$ In doing so, firstly, we establish the fractional Sobolev/ logarithmic Sobolev/ Hardy trace inequalities in terms of $\nabla_{(x,t)}u(x,t).$ Then, we prove the fractional anisotropic Sobolev/ logarithmic Sobolev/ Hardy trace inequalities in terms of $ {\partial_{t} u(x,t)}$ or $(-Δ)^{-γ/2}u(x,t)$ only. Moreover, based on an estimate of the Fourier transform of the Caffarelli…
Large-time asymptotics for degenerate cross-diffusion population models with volume filling
本文研究有界域中退化抛物线交叉扩散系统的大时间渐近性，允许不同扩散率与退化非线性，通过相对熵方法结合凸Sobolev不等式处理修正熵密度完成证明。
In this manuscript we study the following optimization problem: given a bounded and regular domain $Ω\subset \mathbb{R}^N$ we look for an optimal shape for the "$\mathrm{W}-$vanishing window" on the boundary with prescribed measure over all admissible profiles in the framework of the Orlicz-Sobolev spaces associated to constant for the "Sobolev trace embedding". In this direction, we establish existence of minimizer profiles and optimal sets, as well as we obtain further properties for such extremals. Finally, we also place special emphasis on analyzing the corresponding optimization problem i…
Sharp Makai-type inequalities for the best Poincaré-Sobolev constants
本文通过新的几何最优界，证明了Poincaré-Sobolev常数可由周长与体积的比值的p次方下界限制，扩展了Makai关于扭应力的旧结果。
For $n\geq 2$, $p\in(1,n)$, the "best $p$-Sobolev inequality" on an open set $Ω\subset\mathbb{R}^n$ is identified with a family $Φ_Ω$ of variational problems with critical volume and trace constraints. When $Ω$ is bounded we prove: (i) for every $n$ and $p$, the existence of generalized minimizers that have at most one boundary concentration point, and: (ii) for $n> 2\,p$, the existence of (classical) minimizers. We then establish rigidity results for the comparison theorem "balls have the worst best Sobolev inequalities" by the first named author and Villani, thus giving the first affirmative…
The mapping properties of fractional derivatives in weighted fractional Sobolev space
本文利用Calderón–Lions插值定理，证明了Riemann–Liouville和Marchaud分数阶导数在加权分数阶Sobolev空间中的具体映射关系。
We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank--Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.
In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for…
By a result of John Ball (1981), a locally orientation preserving Sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension. As shown here for any dimension, the conclusions of Ball's theorem and related results can be reached while completely avoiding the problem of homeomorphic extension. For suitable domains, it is enough to know that the trace is invertible on the boundary or can be uniformly approximated by such maps. An application in Nonlinear Elasticity is the existence of homeomorphic minimizers with finite distortion whose bounda…
We consider a variant of the classic Steklov eigenvalue problem, which arises in the study of the best trace constant for functions in Sobolev space. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically upon variation of the underlying domain and we compute the corresponding Hadamard-type formulas for the shape derivatives. We also consider isovolumetric and isoperimetric domain perturbations and we characterize the corresponding critical domains in terms of appropriate overdetermined systems. Finally, we prove that balls are critical domains for the ele…
Log-Sobolev inequality for the $φ^4_2$ and $φ^4_3$ measures
本文证明在磁化率有界假设下，连续$φ^4_2$和$φ^4_3$测度在格点正则化中一致满足对数Sobolev不等式，其证明结合了Polchinski重正化群方程准则、Ising模型相关不等式及骨架不等式。
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