Pekka Pankka, Pietro Poggi-Corradini, Kai Rajala
We use geometric methods to obtain a sharp exponential integrability result for boundary traces of monotone Sobolev functions defined on the unit ball.
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We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded $1$-dimensional topological submanifold of the plane. Here monotonicity is in the sense of Lebesgue: the maximum and minimum of the function in an open set are attained at the boundary. Our result is an analog of Sard's theorem, which asserts that for a $C^2$-smooth function in a planar domain almost every value is a regular va…
We extend a Poincaré-type inequality for functions with large zero-sets by Jiang and Lin to fractional Sobolev spaces. As a consequence, we obtain a Hausdorff dimension estimate on the size of zero sets for fractional Sobolev functions whose inverse is integrable. Also, for a suboptimal Hausdorff dimension estimate, we give a completely elementary proof based on a pointwise Poincaré-style inequality.
In this paper, we prove the decay estimates of the velocity and $H^1$ scattering for the 2D linearized Euler equations around a class of monotone shear flow in a finite channel. Our result is consistent with the decay rate predicted by Case in 1960.
We apply the metrical approach to Sobolev spaces, which arise in various evolution PDEs. Functions from those spaces are defined on an interval and take values in a family of Banach spaces. In this case we adapt the definition of Newtonian spaces. For a monotone family, we show the existence of weak derivative, obtain an isomorphism to the standard Sobolev space, and provide some scalar characteristics.
The author establishes some geometric criteria for a domain of ${\mathbb R}^n$ with $n\ge2$ to support a $(pn/(n-ps),\,p)_s$-Hajłasz-Sobolev-Poincaré imbedding with $s\in(0,\,1]$ and $p\in(n/(n+s),\,n/s)$ or an $s$-Hajłasz-Trudinger imbedding with $s\in(0,\,1]$.
Leray-Trudinger Type Exponential Integrability in Log-Weighted Sobolev Spaces
本文研究了具有对数权重的加权索博列夫空间，扩展了勒雷能量与非径向函数的联系，建立了通用函数的最优指数积分性，并证明了径向函数的最优不等式。
We consider the dispersive Degasperis-Procesi equation $u_t-u_{x x t}-\mathtt{c} u_{xxx}+4 \mathtt{c} u_x-u u_{xxx}-3 u_x u_{xx}+4 u u_x=0$ with $\mathtt{c}\neq 0$. In \cite{Deg} the authors proved that this equation possesses infinitely many conserved quantities. We prove that, in a neighborhood of the origin, there are infinitely many of such constants of motion which control the Sobolev norms and which are analytic in a neighborhood of the origin of some $H^s$ Sobolev space, both on $\mathbb{R}$ and $\mathbb{T}$. By the analysis of these conserved quantities we deduce a result of global wel…
Transition Threshold for Strictly Monotone Shear Flows in Sobolev Spaces
针对二维Navier-Stokes方程中带小粘度的谱稳定严格单调剪切流，建立了$s \geq 2$时$H^s$空间的非线性稳定性（阈值为$\varepsilon \nu^{1/3}$，时间小于$c_* \nu^{-1}$），并证明了非线性无粘阻尼和增强耗散。
Asymptotic analysis of fractional Sobolev spaces on thin films in the low-integrability regime
研究薄片上分数Sobolev空间的渐进行为，证明其趋近于H^{s+1/2}(ω)空间，通过维度缩减收敛概念和适当缩放的Gagliardo半范数定义等 coercive 函数。
Involutivity is a well known necessary condition for integrability of smooth tangent distributions. We show that this condition is still necessary for integrability with Sobolev surfaces. We specialize our study to the left invariant horizontal distribution of the first Heisenberg group $\H^1$. Here we answer a question raised in a paper by Z.M.Balogh, R.Hoefer-Isenegger, J.T.Tyson.
By using quasi-Banach techniques as key ingredient we prove Poincaré- and Sobolev- type inequalities for $m$-subharmonic functions with finite $(p,m)$-energy. A consequence of the Sobolev type inequality is a partial confirmation of Błocki's integrability conjecture for $m$-subharmonic functions.
In a previous article, \cite{Zill3}, we have established linear inviscid damping for a large class of monotone shear flows in a finite periodic channel and have further shown that boundary effects asymptotically lead to the formation of singularities of derivatives of the solution. As the main results of this article, we provide a detailed description of the singularity formation and establish stability in all sub-critical fractional Sobolev spaces and blow-up in all super-critical spaces. Furthermore, we discuss the implications of the blow-up to the problem of nonlinear inviscid damping in a…
We prove density of smooth functions in subspaces of Sobolev- and higher order $BV$-spaces of kind $W^{m,p}(Ω)\cap L^q(Ω-D)$ and $BV^m(Ω)\cap L^q(Ω-D)$, respectively, where $Ω\subset\mathbb{R}^n$ ($n\in\mathbb{N}$) is an open and bounded set with suitably smooth boundary, $m<n$ is a positive integer, $1\leq p<\infty$ s.t. $mp<n$, $D\SubsetΩ$ is a sufficiently regular open subset and $q> np/(n-mp)$. Here we say that a $W^{m-1,1}(Ω)$-function is of $m$-th order bounded variation ($BV^m$) if its $m$-th order partial derivatives in the sense of distributions are finite Radon measures. This takes u…
This work is concerned with both higher integrability and differentiability for linear nonlocal equations with possibly very irregular coefficients of VMO-type or even coefficients that are merely small in BMO. In particular, such coefficients might be discontinuous. While for corresponding local elliptic equations with VMO coefficients such a gain of Sobolev regularity along the differentiability scale is unattainable, it was already observed in previous works that gaining differentiability in our nonlocal setting is possible under less restrictive assumptions than in the local setting. In th…
Let $d \ge 1$, $p \ge d$, and let $Ω$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $Ω$ such that $$ \mathop{\int_Ω \int_Ω}_{|u(x) - u(y)| > δ} \frac{1}{|x-y|^{d+p}} \, dx \, dy < + \infty, $$ for some $δ> 0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincaré inequalities, and Hardy inequalities.
We give a simple proof of the $λ= d-2$ cases of the sharp Hardy-Littlewood-Sobolev inequality for $d\geq 3$, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for $d=2$ via a monotone flow governed by the fast diffusion equation.
We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations
Monotone Sobolev functions: approximation, critical points, and level sets
该论文研究单调Sobolev函数的逼近、临界点与水平集，给出平面局部光滑性的肯定答案及高维问题部分肯定与否定答案，提出光滑单调逼近并应用于多个领域。
Generalized Newton-Leibniz Formula and the Embedding of the Sobolev Functions with Dominating Mixed Smoothness into Hölder Spaces
本文推广Newton-Leibniz公式至n维矩形，结合Sobolev迹嵌入归纳证明，给出与可积指数无关的弱可微条件，刻画Sobolev函数到Hölder空间的嵌入，并证明带主导混合光滑性的Sobolev空间的对应嵌入结论。
Kinetic Sobolev Spaces
本文定义并研究了适应Kolmogorov方程的同源动能Sobolev空间，考虑了局部和非局部扩散，通过Lebesgue空间L^p和Kolmogorov方程的尺度建立正则性假设，证明了这些空间满足精确的嵌入性质，包括正则性转移、时间连续性和积分性增益。
We prove trace theorems for weighted mixed norm Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They are new even in the un-weighted case and they recover classical results in the case of un-mixed norm spaces. The work is motivated by the study of re…
Sharp defective log-Sobolev inequalities on H-type groups
本文在H型群上建立尖锐缺陷对数Sobolev不等式，并借此证明Lipschitz函数关于热核测度的指数可积性，同时给出高斯型测度关于次黎曼距离的缺陷对数Sobolev型不等式。
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.
In this paper, we revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity $L^1_t W^{1,p}$ for all $p<\infty$ in space dimensions $d\geq 2$ whose transport equations admit nonunique weak solutions belonging to $L^p_tC^k$ for all $p<\infty$ and $k\in \mathbb{N}$. In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna-Lions theory is sharp. The same result also holds for transport-diffusion equations with diffusion ope…
On Choquet integrals and Sobolev type inequalities
本文研究n维欧氏空间开连通集上连续可微函数关于δ维Hausdorff容度的Choquet积分，证明了p=δ/n极限情形与p>δ情形的Sobolev型不等式，补充了已有的Poincaré-Sobolev和Morrey不等式。
Global invertibility of Sobolev mappings with prescribed homeomorphic boundary values
研究具有正雅可比行列式的Sobolev映射在给定同胚边界条件下的整体可逆性，证明该类映射连续延拓到闭包且是单调连续满射，推广了Ball的开创性工作。
In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue space, under various integrability regimes. Our method is based on an integral refinement of the Bochner's identity, and leads to "semilinear Calderón-Zygmund" type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclide…
We derive sharp Adams inequalities for the Riesz and more general Riesz-like potentials on the whole of R^n. As a consequence, we obtain sharp Moser-Trudinger inequalities for the critical Sobolev spaces W^{a,n/a}(R^n), 0<a<n. These inequalities involve fractional Laplacians, higher order gradients, general homogeneous elliptic operators with constant coefficients, and general trace type Borel measures.
Homogeneous Newton-Sobolev spaces in metric measure spaces and their Banach space properties
本文在配备加倍测度且支撑 p-Poincaré 不等式的无界度量测度空间上，证明了齐次 Newton-Sobolev 空间 HN^{1,p}(X) 的 Banach 空间性质，并证明当 1<p<∞ 时，即使缺乏全局 L^p 可积性，该空间中的每个有界序列都有强收敛的凸组合子序列。
We study topologically monotone surjective $W^{1,n}$-maps of finite distortion $f \colon Ω\to Ω'$, where $Ω, Ω' $ are domains in $\mathbb{R}^n$, $n \geq 2$. If the outer distortion function $K_f \in L_{\mathrm{loc}}^{p}(Ω)$ with $p \geq n-1$, then any such map $f$ is known to be homeomorphic, and hence the fibers $f^{-1}\{y\}$ are singletons. We show that as the exponent of integrability $p$ of the distortion function $K_f$ increases in the range $1/(n-1) \leq p < n-1$, then the fibers $f^{-1}\{y\}$ of $f$ start satisfying increasingly strong homological limitations. We also give a Sobolev rea…
Fractional Sobolev spaces with kernel function on compact Riemannian manifolds
本文提出了一类紧致黎曼流形上带核函数的分数阶 Sobolev 空间，证明了其可分性、自反性、完备性及嵌入结果，并利用拓扑度理论证明了涉及分数阶 $p(\cdot, \cdot)$-Laplacian 算子的非局部椭圆问题解的存在性。
We prove the Lewy-Stampacchia inequalities for the two obstacles problem in abstract form for T-monotone operators. As a consequence for a general class of quasi-linear elliptic operators of Ladyzhenskaya-Uraltseva type, including p(x)-Laplacian type operators, we derive new results of $C^{1,α}$ regularity for the solution. We also apply those inequalities to obtain new results to the N-membranes problem and the regularity and monotonicity properties to obtain the existence of a solution to a quasi-variational problem in (generalized) Orlicz-Sobolev spaces.
Distortion of Hausdorff measures under Orlicz--Sobolev maps
本文系统研究了 Orlicz-Sobolev 映射对零或有限 Hausdorff 测度子集的影响，用 Young 函数给出规范函数畸变的显式公式，并发现新现象、恢复并强化了经典 Sobolev 空间结果。
Anisotropic Improved Leray-Trudinger Inequality
本文在强凸Finsler范数诱导的各向异性Sobolev空间中建立了Leray-Trudinger型不等式，推广了经典指数可积性结果，并在各向异性径向函数类中得到了Moser意义下的最优常数。
We study extremal functions for a family of Poincaré-Sobolev-type inequalities. These functions minimize, for subcritical or critical $p\geq 2$, the quotient ${\|\nabla u\|_2}/{\|u\|_p}$ among all $u \in H^1(B)\setminus\{0\}$ with $\int_{B}u=0$. Here $B$ is the unit ball in $\mathbb{R}^N$. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of $B$. We also prove that, for…
Velocity-vorticity geometric constraints for the energy conservation of 3D ideal incompressible fluids
本文针对三维Euler方程，提出结合空间分数阶Sobolev正则性与时间可积性的能量守恒判据，并借助Sobolev空间乘积性质与bootstrap论证，将其应用于Beltrami型解的能量守恒研究。
It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality \[\mathcal{H}^β_{\infty}(\{x\in Ω:|I_αf(x)|>t\})\leq Ce^{-ct^{q'}}\] for all $\|f\|_{L^{N/α,q}(Ω)}\leq 1$ and any $β\in (0,N]$, where $Ω…
Unified theory for regularity persistence of vortex patch boundaries
针对含径向卷积核的二维有源标量方程，建立涡斑边界 Sobolev 正则性保持的统一局部理论，证明特定条件下轮廓动力学方程的局部解存在唯一性，丰富了这类方程的正则性研究成果。
Deficit functions and the log Sobolev inequality
本文引入椭圆型与抛物型两个亏量函数，证明后者是前者的高维逐点极限且方程相应收敛，从而恢复单调性并给出对数 Sobolev 不等式的椭圆型证明。
Sobolev inequalities and regularity of the linearized complex Monge-Ampere and Hessian equations
本文研究复$k$-Hessian上下有界时线性化复Monge-Ampère与Hessian方程的正则性，通过建立Green函数估计和新Sobolev不等式，利用Moser迭代得到先验估计与Harnack不等式，适用于实与复情形。
Local convergence of mean-field Langevin dynamics: from gradient flows to linearly monotone games
本文研究了均场动力学的局部收敛性，通过梯度流和线性单调博弈的分析，提出了在χ²-分歧下具有精确收敛速度的局部指数收敛性结果。
We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any $3\le n$-dimensional complete non-compact boundary-free Riemannian manifold through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough and sharp Euclidean isoperimetric inequalities. Consequently, we are led to evaluate the critical limit of an induced monotone Green's functional using the asymptotic behavior of the Lorentz norm deficit of Green's function at the infinity, as well as the harmonic radius of a regular …
For kernels $ν$ which are positive and integrable we show that the operator $g\mapsto J_νg=\int_0^x ν(x-s)g(s)ds$ on a finite time interval enjoys a regularizing effect when applied to Hölder continuous and Lebesgue functions and a "contractive" effect when applied to Sobolev functions. For Hölder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor $N(x)=\int_0^x ν(s)ds$. For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of…
Effective dynamics and defect expansions for polynomial PDEs on thin annuli
本文提出了一种基于Sobolev正交多项式的方法，用于分析薄环区域中多项式偏微分方程的有效动力学和缺陷展开，适用于多种物理模型和扰动情况。
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