Dominic Breit, Lars Diening, Franz Gmeineder
We construct a Lipschitz truncation which approximates functions of bounded variation in the area-strict metric. The Lipschitz truncation changes the original function only on a small set similar to Lusin's theorem. Previous results could only give estimates on the Lebesgue measure of the set where the Lipschitz approximations differ from the original function.
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We develop an improved version of the parabolic Lipschitz truncation, which allows qualitative control of the distributional time derivative and the preservation of zero boundary values. As a consequence, we establish a new caloric approximation lemma. We show functions. The distance is measured in terms of spatial gradients as well as almost uniformly in time. Both results are extended to the setting of Orlicz growth.
Lipschitz truncation method for parabolic double-phase systems and applications
本文提出一种基于 Whitney 型覆盖和单位分解的 Lipschitz 截断方法，用于证明 $p$-Laplace 型抛物双相 Dirichlet 问题的能量估计与唯一性，并展示非齐次问题的存在性。
Parabolic Lipschitz truncation for multi-phase problems: the degenerate case
本文研究抛物型多相问题中的Lipschitz截断方法，基于Whitney分解和覆盖引理，并利用文献[24]中的比较方案区分各相，处理退化情形。
An alternative approach to solenoidal Lipschitz truncation
本文提出了一种新的方法，用于获得无旋Lipschitz截断，通过严格控制W^{1,p}距离来改进传统方法。
Solenoidal Lipschitz truncation for parabolic PDE's
我们考虑函数$u\in L^\infty(L^2)\cap L^p(W^{1,p})$与$1<p<\infty$的时间空间域上。非线性演化偏微分方程的解通常属于这些空间。许多应用需要$u$的Lipschitz近似$u_λ$，其在大集合上与$u$一致。对于流体力学中出现的问题，需要使用螺线管（无发散）函数。因此，我们构造了一个Lipschitz近似，它也是螺线管。作为应用，我们修正了文献[DRW 10]中关于非定常广义牛顿流体存在性的证明。由于${\rm div} u_λ=0$，我们能够在无压力的配方，大大简化了证明工作。我们还提供了一个简化的方法来固定螺线管Lipschitz截断[BDF 12]。
Existence of mild solutions for the 3D MHD system in bounded Lipschitz domains is established in critical spaces with the absolute boundary conditions.
Boundary integral operator for the fractional Laplace equation in a bounded Lipschitz domain
研究有界Lipschitz域上分数拉普拉斯方程诱导的边界积分算子。作为应用，我们研究了一个分数阶Laplace方程的边值问题.
Lipschitz regularity for a priori bounded minimizers of integral functionals with nonstandard growth
该研究在增长指数与椭圆性指数之差满足严格界的条件下，建立了满足非标准增长条件的非自治积分泛函先验有界局部极小元的Lipschitz正则性。
We prove local Lipschitz regularity for bounded minimizers of functionals with nonstandard $p,q$-growth with the source term in the Lorentz space $L(N,1)$ under the restriction $q<p+1+p\,\min\left\{\frac 1N,\frac{2(p-1)}{Np-2p+2}\right\}$. This extends the recent work by Beck-Mingione to bounded minimizers under weaker hypothesis and is sharp for some special ranges of $p$, $q$ and $N$.
In this work we prove some abstract results about the existence of a minimizer for locally Lipschitz functionals, without any assumption of homogeneity, over a set which has its definition inspired in the Nehari manifold. As applications we present a result of existence of ground state bounded variation solutions of problems involving the 1-Laplacian and the mean-curvature operator, where the nonlinearity satisfies mild assumptions.
A Dirichlet problem of the fractional Laplace equation in the bounded Lipschitz domain
本文研究了$ \R, n \geq 2$中有界Lipschitz区域上分数阶Laplace方程的一个狄利克雷问题.我们的主要结果是，对于给定的数据$F \in \dot H^s(\Om^c), 0 < s<1$，我们找到了满足$\Om$，$u|_{\Om^c} =F$和$|u|_{\dot{H}^s(\R)} \leq c |F|_{\dot H^s(\Om^c)}$中的$\De^s u =0$的函数。此外，我们用一个积分算子表示解。
Global Lipschitz and Sobolev estimates for the Monge-Ampère eigenfunctions of general bounded convex domains
本文证明了一般有界凸区域上Monge-Ampère特征函数及一类退化Monge-Ampère方程凸解的全局Lipschitz性质，并由此导出了全局$W^{2,1}$估计。
Local Lipschitz continuity of the minimizers of nonuniformly convex functionals under the Lower Bounded Slope Condition
本文在$g$有界且$ϕ$满足下有界斜率条件的情况下，证明了非一致凸泛函极小值点的局部Lipschitz正则性，并作为副产品证明了允许$f$非凸时该类泛函局部Lipschitz极小值点的存在性。
We use the scale $B^s_τ(L_τ(Ω))$, $1/τ=s/d+1/2$, $s>0$, to study the regularity of the stationary Stokes equation on bounded Lipschitz domains $Ω\subset\mathbb{R}^d$, $d\geq 3$, with connected boundary. The regularity in these Besov spaces determines the order of convergence of nonlinear approximation schemes. Our proofs rely on a combination of weighted Sobolev estimates and wavelet characterizations of Besov spaces. By using Banach's fixed point theorem, we extend this analysis to the stationary Navier-Stokes equation with suitable Reynolds number and data, respectively.
On fractional Orlicz boundary Hardy inequalities
本文研究有界Lipschitz域上的分数阶Orlicz边界Hardy型不等式，并在多类域中对特定临界情形建立了带对数修正的不等式。
Global wellposedness to incompressible inhomogeneous fluid system with bounded density and non-Lipschitz velocity
本文首先证明了临界Besov空间中d维不可压缩非齐次Navier-斯托克斯方程在满足非线性小性条件下整体弱解的存在性.初始速度的正则性对系统的标度性至关重要，并且足够一般以产生非Lipschitz速度场。此外，通过对初始速度或初始密度的附加正则性假设，我们也可以证明这种解的唯一性。我们应该提到，经典的最大正则性定理的热核在这方面起着至关重要的作用。
The paper extends the results obtained by C. Kenig, F. Lin and Z. Shen in \cite{SZW2} to more general elliptic homogenization problems in two perspectives: lower order terms in the operator and no smoothness on the coefficients. We do not repeat their arguments. Instead we find the new weighted-type estimates for the smoothing operator at scale $\varepsilon$, and combining some techniques developed by Z. Shen in \cite{SZW12} leads to our main results. In addition, we also obtain sharp $O(\varepsilon)$ convergence rates in $L^{p}$ with $p=2d/(d-1)$, which were originally established by Z. Shen …
We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard $(p,q)$-growth satisfying the dimension-independent restriction $q<p+2$ with $p \geq 2$. This relation improves existing restrictions when $p \leq N-1$, moreover our result is sharp in the range $N > \frac{p(2+p)}{2} + 1$. The standard Lipschitz regularity takes the form $W^{1,\infty}_{\text{loc}} - W^{1,p}_{\text{loc}}$, whereas we obtain $W^{1,\infty}_{\text{loc}} - L^{\infty}_{\text{loc}}$ regularity estimate and then make use of existing sharp $L^{\infty}_{\text{loc}}$ bounds to obtain the required conclus…
Robin-to-Robin Maps and Krein-Type Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains
本文研究了$\bbR^n$，$n\ge 2$中有界Lipschitz域上的Robin-to-Robin映射和薛定谔算子在广义Robin边界条件下的Krein型预解式.
In this paper we study the existence of weak solutions to an unsteady system describing the motion of micro-polar electrorheological fluids. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. Using the Lipschitz truncation and the solenoidal Lipschitz truncation we establish the existence of global solutions for shear exponents $p>6/5$ in three-dimensional domains.
Lipschitz stability for an elliptic inverse problem with two measurements
本文研究通过包围区域边界上的测量值反演椭圆方程外部未知边界值的问题，在可容许函数类上证明了Lipschitz稳定性，并将结果推广到内部问题和抛物情形。
Generalized Robin Boundary Conditions, Robin-to-Dirichlet Maps, and Krein-Type Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains
本文研究了$\bbR^n$，$n\ge 2$中有界Lipschitz域上薛定谔算子的广义Robin边界条件，Robin-to-Dirichlet映射和Krein型预解式.我们还讨论了有界$C^{1,r}$ -整环$(1/2)<r<1$的情形.
A Characterization of Solvability of the Parabolic $L^p$ Dirichlet Problem on Lipschitz Graph Domains Via Carleson Measure Estimates of Bounded Solutions
该研究通过有界解的Carleson测度估计，刻画了Lipschitz图区域上抛物型$L^p$Dirichlet问题的可解性，改进了需附加抛物一致可求长性等额外假设的现有结果。
In this work we establish solvability and uniqueness for the $D_2$ Dirichlet problem and the $R_2$ Regularity problem for second order elliptic operators $L=-{\rm div}(A\nabla\cdot)+b\nabla\cdot$ in bounded Lipschitz domains, where $b$ is bounded, as well as their adjoint operators $L^t=-{\rm div}(A^t\nabla\cdot)-{\rm div}(b\,\cdot)$. The methods that we use are estimates on harmonic measure, and the method of layer potentials. The nature of our techniques applied to $D_2$ for $L$ and $R_2$ for $L^t$ leads us to impose a specific size condition on ${\rm div}b$ in order to obtain solvability. O…
Resolvent Estimates for the Stokes Operator in a Three-Dimensional Lipschitz Domain
通过改进已有方法，建立三维Lipschitz区域Stokes算子的L^p预解估计，证明其生成一致有界解析半群，补充了插值相关定理，修正了大p下估计可能失效的传统认知。
Calderon-Zygmund estimates for stochastic elliptic systems on bounded Lipschitz domains
本文研究有界 Lipschitz 区域上具可积相关随机系数椭圆系统的 Calderón-Zygmund 估计，并借助新型最小半径与 Gloria-Neukamm-Otto 策略得到近乎最优的振荡与涨落均匀化误差。
We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain $Ω$ subject to homogeneous Dirichlet boundary conditions. We prove $\mathrm{L}^p$-resolvent estimates for $p$ satisfying the condition $\lvert 1 / p - 1 / 2 \rvert < 1 / 4 + \varepsilon$ for some $\varepsilon > 0$. We further show that the Stokes operator admits the property of maximal regularity and that its $\mathrm{H}^{\infty}$-calculus is bounded. This is then used to characterize domains of fractional powers of the Stokes operator. Finally, we give an application to the regularity theory of weak solutio…
Heat Kernel Estimates for Schrödinger Operators in the Domain Above a Bounded Lipschitz Function
本文研究有界 Lipschitz 图像上方区域内 Schrödinger 算子的 Dirichlet 热核，并在势函数远离边界呈超二次衰减时建立匹配的上下界。
We prove that for bounded Lipschitz domains in $\mathbb{R}^N$ Korn's first inequality holds for vector fields satisfying homogeneous mixed normal and tangential boundary conditions.
In this paper we study the resolvent of wave operators on open and bounded Lipschitz domains of $\mathbb{R}^N$ with Dirichlet or Neumann boundary conditions. We give results on existence and estimates of the resolvent for the real and complex cases.
Estimates on the stability constant for the truncated Fourier transform
该文研究截断傅里叶变换恢复紧支撑函数的反问题，推导了关于截断参数的Lipschitz稳定性估计，揭示常数的量级变化规律，并通过含噪重构数值实验验证了理论结果。
In this note we show that gradient of Harmonic functions on a smooth domain with Lipschitz boundary values is pointwise bounded by a universal function which is in $L^p$ for all finite $p\geq 1$.
Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance
最近发展的理论框架，分析结构人口动力学的非负氡措施的空间与适当的度量提供了一个严格的工具，研究基于粒子方法的数值方案。该方法是基于跟踪逼近原偏微分方程解的测度的增长和迁移的思想。在本文中，我们提出了两个版本的自动扶梯Boxcar训练（EBT）算法，已被广泛应用于理论生物学的分析和数值研究，并比较它最近开发的分裂算法。本文的新颖之处在于利用度量空间上半流的概念证明了该方案的适定性和收敛速度。理论结果通过测试用例的数值模拟进行了验证，其中使用平坦度量计算模拟解与精确解之间的距离。
We consider the functional $\int_Ωg(\nabla u+\textbf X^\ast)d\mathscr L^{2n}$ where $g$ is convex and $\textbf X^\ast(x,y)=2(-y,x)$ and we study the minimizers in $BV(Ω)$ of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on $g$, there exists a unique minimizer which is also Lipschitz continuous.
We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional are locally Lipschitz continuous functions, without any restriction on the anisotropy.
Let $\mathcal{B}$ be a homogeneous differential operator of order $l=1$ or $l=2$. We show that a sequence of functions of the form $(\mathcal{B}u_j)_j$ converging in the $L^1$-sense to a compact, convex set $K$ can be modified into a sequence converging uniformly to this set provided that the derivatives of order $l$ are uniformly bounded. We prove versions of our result on the whole space, an open domain, and for $K$ varying uniformly continuously on an open, bounded domain. This is a conditional generalization of a theorem proved by S. Müller for sequences of gradients, cf. [6]. Moreover, a …
Given a semi-convex potential V on a convex and bounded domain $Ω$, we consider the Jordan-Kinderlehrer-Otto scheme for the Fokker-Planck equation with potential V, which defines, for fixed time step $τ$ > 0, a sequence of densities $ρ$ k $\in$ P($Ω$). Supposing that V is $α$-convex, i.e. D 2 V $\ge$ $α$I, we prove that the Lipschitz constant of log $ρ$ + V satisfies the following inequality: Lip(log($ρ$ k+1) + V)(1 + $α$$τ$) $\le$ Lip(log($ρ$ k) + V). This provides exponential decay if $α$ > 0, Lipschitz bounds on bounded intervals of time, which is coherent with the results on the continuous…
Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains
本文在有界Lipschitz域中利用临界Besov空间和最优$\mathrm{L}^q$-正则性，证明了含一般耦合项的Keller-Segel-Navier-Stokes系统在小初值下存在局部与全局强解，并得到平衡态指数稳定性及解的正性保持。
We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows "forbidden" endpoints and permits the consideration of functions exhibiting very limited regularity. This is done at the (necessary) expense of stipulating an additional regularity condition involving the action of the Laplacian on the functions in question which, nonetheless, works perfectly with the Dirichlet and Neumann realizations of the Schrödinger differential expression $-Δ+V$. In turn, this boundary trace theory serves as a platform for developing a spectral …
On the First Eigenfunction of the Symmetric Stable Process in a Bounded Lipschitz Domain
证明了$\R^d$中有界Lipschitz域上的$α$对称稳定过程的第一特征函数$d\geq 1$对于$α=2/m$是超调和的，其中$m>2$是整数.这个结果首先由M. Kaßmann和L. Silvestre（个人沟通）使用不同的方法。对于$α=1$，在\cite [定理4.7]{BanKul}中证明了该结果。
A Compactness Result for the div-curl System with Inhomogeneous Mixed Boundary Conditions for Bounded Lipschitz Domains and Some Applications
本文证明有界 Lipschitz 区域上非齐次混合边界条件下 div-curl 向量场的 $L^2$ 紧性，并用于建立 Friedrichs/Poincaré 估计、div-curl 引理及 Maxwell 算子紧预解式。
Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
研究加权Morrey空间上Hardy-Littlewood极大函数与sharp极大函数交换子的有界性，并据此给出加权Lipschitz函数的新刻画。
We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the complement of a bounded Lipschitz domain in fractional Orlicz-Sobolev setting. As a consequence, we get sufficient conditions for regional fractional Orlicz Poincaré inequality in bounded Lipschitz domains. Necessary conditions for fractional Orlicz Hardy and regional fractional Orlicz Poincaré inequalities are also given for bounded Lipschitz domains. Various sufficient conditions …
In this paper we study the self-adjoint Krein-von Neumann realization $A_K$ of the perturbed Laplacian $-Δ+V$ in a bounded Lipschitz domain $Ω\subset\mathbb{R}^n$. We provide an explicit and self-contained description of the domain of $A_K$ in terms of Dirichlet and Neumann boundary traces, and we establish a Weyl asymptotic formula for the eigenvalues of $A_K$.
Truncations for fractional Laplacians
该研究证明并推广了Nazarov的猜想，针对s∈(1, 3/2)的情况，当$\tilde H^s(Ω)$中的函数u变号时，谱分数阶狄利克雷拉普拉斯算子的二次型在映射u→|u|下严格增大。
Dirichlet-to-Neumann maps on bounded Lipschitz domains
当一个椭圆型偏微分方程的狄利克雷问题不唯一可解时，它的Dirichlet-to-Neumann映射成为多值映射.本文的主要目的是在希尔伯特空间的线性关系的框架下，系统地研究Dirichlet-to-Neumann映射及其逆映射Neumann-to-Dirichlet映射.我们的处理是从对称算子的扩张理论的抽象方法的启发，利用线性关系的一般理论，并利用一些深刻的结果的正则性的边值问题有界Lipschitz域上。
Conditonal Lipschitz stability for the Inverse Problem of the 2D Navier-Stokes System in a Bounded Domain
本文研究了二维纳维-斯托克斯系统在有限域中的逆问题，通过条件Lipschitz稳定性与局部恢复方法，利用初始速度场和全局涡量不变量恢复局部速度场和边界涡量。
Boundary regularity for anisotropic minimal Lipschitz graphs
本文证明了在各向异性能量满足一致标量原子条件时，具有$C^{1,α}$边界且平均曲率在$L^p$中有界的任意余维数Lipschitz图像在特定密度边界点处的正则性。
For a bounded weak Lipschitz domain we show the so called `Maxwell compactness property', that is, the space of square integrable vector fields having square integrable weak rotation and divergence and satisfying mixed tangential and normal boundary conditions is compactly embedded into the space of square integrable vector fields. We will also prove some canonical applications, such as Maxwell estimates, Helmholtz decompositions and a static solution theory. Furthermore, a Fredholm alternative for the underlying time-harmonic Maxwell problem and all corresponding and related results for exter…
It is proved that the space of differential forms with weak exterior and co-derivative, is compactly embedded into the space of square integrable differential forms. Mixed boundary conditions on weak Lipschitz domains are considered. Furthermore, canonical applications such as Maxwell estimates, Helmholtz decompositions and a static solution theory are proved. As a side product and crucial tool for our proofs we show the existence of regular potentials and regular decompositions as well.
This work provides a geometric characterization of the measures $μ$ in $\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $Rμ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$. More precisely, it is shown that $$\|Rμ\|_{L^2(μ)}^2 + \|μ\|\approx \int\!\!\int_0^\infty β_{2,μ}(x,r)^2\,\frac{μ(B(x,r))}{r^n}\,\frac{dr}r\,dμ(x) + \|μ\|,$$ where $β_{μ,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{dist(y,L)}r\right)^2\,dμ(y),$ with the infimum taken over all affine $n$-planes $L\subset \mathbb R^{n+1}$. As a corollary, one o…
Extrapolation of solvability of the parabolic $L^p$ Neumann problem on bounded Lipschitz cylinders
本文研究了抛物型 $L^p$ 霍尔德问题在有界利普希茨圆柱上的可解性外推问题，提出新的方法以解决该问题。
On a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$, $d \geq 3$, we continue the study of Shen and of Kunstmann and Weis of the Stokes operator on $\mathrm{L}^p_σ (Ω)$. We employ their results in order to determine the domain of the square root of the Stokes operator as the space $\mathrm{W}^{1 , p}_{0 , σ} (Ω)$ for $\lvert \frac{1}{p} - \frac{1}{2} \rvert < \frac{1}{d} + \varepsilon$ and some $\varepsilon > 0$. This characterization provides gradient estimates as well as $\mathrm{L}^p$-$\mathrm{L}^q$-mapping properties of the corresponding semigroup. In the three-dimensional case this provi…
It is shown that, generically, a positive continuous functions cannot be written as the derivative of a bi-Lipschitz plane homeomorphism. The same is proved for generic bounded functions.
Transversally Lipschitz Harmonic Functions are Lipschitz
设Ω \subset \mathbb{R}^n是具有C^ \infty边界的有界域.我们证明了Ω中的一个调和函数，如果它沿着一族横截于bΩ的曲线是Lipschitz的，那么它也是Ω中的Lipschitz.我们考虑的Lipschitz函数空间是使用优数的概念定义的，优数是幂函数t^α，0<α<1的某种推广。
BV Minimizers of the area functional in the Heisenberg group under the bounded slope condition
本文研究了次黎曼海森堡群中t-图的面积泛函，并研究了相应的狄利克雷问题的极小解.证明了在边界数据上的有界斜率条件下，存在唯一的极小元，且此极小元是Lipschitz连续的。我们还提供了一个例子表明，在第一海森堡群，Lipschitz正则性是尖锐的，即使在有界斜率条件。
The Hot Spots constant for bounded smooth domains was recently introduced by Steinerberger (2021) as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We generalize the Hot Spots constant to bounded Lipschitz domains and show that it leads to an if and only if condition for the weak Hot Spots conjecture HS2 from Bañuelos and Burdzy (1999). We also derive a new general formula for a dimension-dependent upper bound that can be tailored to any specific class of domains. This formula is then used to compute upper bounds fo…
A note on a residual subset of Lipschitz functions on metric spaces
设（X，d）是具有参考概率测度m的拟凸完备可分度量空间.证明了在Lipschitz和有界函数的巴拿赫空间中，具有非零点Lipschitz常数m-几乎处处的实值Lipschitz函数的集合是剩余的，因而是稠密的.这个结果是Alberti，Bianchini和Crippa在[1]中对定义在R2上的实值Lipschitz映射所证明的一个结果的度量类似。
Let $n\ge2$ and $Ω$ be a bounded Lipschitz domain in $\mathbb{R}^n$. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in $Ω$. More precisely, let $p\in(n/(n-1),\infty)$. Using a real-variable argument, the authors obtain two necessary and sufficient conditions for $W^{1,p}$ estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent $p$ or …
Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates
本文提出用光滑有界区域列逼近Lipschitz区域的新方法，支持内、外逼近，适用于边界具有弱曲率（局部为$W^{2,q}$函数图像）的区域，并建立关于初始区域的一致等容估计。
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