Carleson Perturbations for the Regularity Problem

Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity

Let $Ω$ be an open set with Ahlfors-David regular boundary satisfying the corkscrew condition. When $Ω$ is connected in some quantitative form one can establish that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting these equivalent properties are stable under Fefferman-Kenig-Pipher perturbations. However, without connectivity, there is no Fefferman-Kenig-Pipher perturbation result available. In this paper, …

The $L^p$ Dirichlet and Regularity problems for second order Elliptic Systems with application to the Lamé system

In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel aspect of our result is that it applies to operators with coefficients of limited regularity and applies to operators satisfying a natural Carleson condition that has been first considered in the scalar case. In this paper we extend this result in several directions. We improve the range of solvability of the $L^p$ Dirichlet problem to the interval $2-\varepsil…

Causal sparse domination of Beurling maximal regularity operators

We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, defined by the $L_1$ Carleson functional and $L_q$ ($1<q<\infty$) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.

Solvability of the Poisson-Dirichlet problem with interior data in $L^{p'}$-Carleson spaces and its applications to the $L^{p}$-regularity problem

We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problem for an elliptic operator $L=-\operatorname{div}A\nabla$ with real and merely bounded coefficients is equivalent to the $L^{p'}$-solvability of the Poisson Dirichlet problem $Lw=H-\operatorname{div} F$, which is defined in terms of an $L^{p'}$ estimate on the non-tangential maximal function, assuming that $\operatorname{dist}(\cdot, \partial Ω) H$ and $F$ lie in certain $L^{p'}$-Carleson-type spaces, and that the domain $Ω\subset\mathbb R^{n+1}$, $n\geq2$, satisfies the corkscrew condition and has $n$-Ahlfors regular bo…

满足小Carleson测度范数条件的复$L^{\infty}$系数椭圆型方程的层势和边值问题

Layer potentials and boundary value problems for elliptic equations with complex $L^{\infty}$ coefficients satisfying the small Carleson measure norm condition

考虑半空间$\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$上的散度型椭圆型方程$Lu:=\nabla\cdot(A\nabla u)=0$，其系数矩阵$A$是复椭圆的、有界的、可测的.此外，我们还假设$A$在横截边界的方向上满足一些额外的正则性，即$A(x,t) -A(x,0)$满足Fundaman-Kenig-Pipher型Carleson测度条件，且Carleson范数较小.在这些条件下，我们建立了$L^p$，Hardy，Sobolev，BMO和Hölder空间中双层和单层势的全范围有界性结果。此外，我们证明$L$的狄利克雷问题的可解性，与数据在$L^p(\mathbb{R}^n)$，$BMO(\mathbb{R}^n)$，和$C^α(\mathbb{R}^n)$，和诺依曼和正则性问题的可解性，与数据分别在空间$L^p(\mathbb{R}^n)/H^p(\mathbb{R}^n)$和$L^p_1(\mathbb{R}^n)/H^{1,p}(\mathbb{R}^n)$，与适当的限制指数，假设可逆性层电位的$t$ -独立运营商$L_0:= -\nabla\cdot(A(\cdot,0)\nabla)$。

二阶复系数椭圆算子解的扰动理论与$L^p$狄利克雷问题

Perturbation theory for solutions to second order elliptic operators with complex coefficients and the $L^p$ Dirichlet problem

对复系数二阶发散型椭圆算子建立了一个Dahlberg型扰动定理。在我们的前一篇文章中，我们证明了如下结果：如果${\mathcal L}_0=\mbox{div} A^0(x)\nabla+B^0(x)\cdot\nabla$是一个满足关于$\nabla A$和$B$的Carleson条件的$p$ -椭圆算子，则算子${\mathcal L}_0$的$L^p$狄利克雷问题在上半空间${\mathbb R}^n_+$中可解.本文证明了在${\mathcal L}_0$的小扰动下$L^p$的可解性是稳定的。也就是说，如果${\mathcal L}_1$是另一个具有复系数的发散型椭圆算子，并且算子${\mathcal L}_0$和${\mathcal L}_1$的系数在Carleson测度意义下（考虑系数的差异）足够接近，则对于相同的$p$值，算子${\mathcal L}_1$的$L^p$狄利克雷问题是可解的.作为推论，我们得到一个新的结果$L^p$可解性的狄利克雷问题的运营商的形式${\mathcal L}=\mbox{div} A(x)\nabla+B(x)\cdot\nabla$的矩阵$A$满足较弱的Carleson条件比我们以前的文件，特别是系数的$A$不再需要是可微的，而是满足Carleson条件，控制振荡的矩阵$A$的惠特尼盒。这一结果在真实情况下已经由Dindobel、P…

Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates

We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation $Lu=-{\rm div}(A\nabla u) = 0$ with $A$ being a real (not necessarily symmetric) uniformly elliptic matrix, imply that the corresponding elliptic measure belongs to the Muckenhoupt $A_\infty$ class…

关于抛物算子的 $L^p$ 正则性问题

The $L^p$ regularity problem for parabolic operators

本文研究了抛物PDE在Lipschitz圆柱上的正则性问题，证明了在某些$p\in(1,\infty)$范围内存在解，并解决了Carleson条件下的问题。

抛物型正则性与Neumann问题的扰动理论

Perturbation theory for the parabolic Regularity and Neumann problem

该研究针对抛物型正则性边值问题与Neumann问题，在Lipschitz区域对应的抛物柱体域内，针对算子$L=\u2202_t-div(A\u2207\u00b7)$，推导了不同边界数据空间下的Carleson扰动结果。

Boundary value problems for elliptic operators satisfying Carleson condition

In this paper we present in concise form recent results, with illustrative proofs, on solvability of the $L^p$ Dirichlet, Regularity and Neumann problems for scalar elliptic equations on Lipschitz domains with coefficients satisfying a variety of Carleson conditions. More precisely, with $L=\mbox{div}(A\nabla)$, we assume the matrix $A$ is elliptic and satisfies a natural Carleson condition either in the form that ($|\nabla A(X)|\lesssim \mbox{dist}(X,\partialΩ)^{-1}$ and $|\nabla A|(X)^2\mbox{dist}(X,\partialΩ)\,dX$) or $\mbox{dist}(X,\partialΩ)^{-1}\left(\mbox{osc}_{B(X,δ(X)/2)}A\right)^2\,d…

$\u211d^n_+$上Dahlberg-Kenig-Pipher算子的$L^p$正则性问题的另一种证明

An alternative proof of the $L^p$-regularity problem for Dahlberg-Kenig-Pipher operators on $\mathbb R^n_+$

本文给出了$\u211d^n_+$上满足Carleson测度条件的一致椭圆算子正则性问题可解性的更简洁替代证明，扩展了可解算子类别，并在$\u211d^n \setminus \u211d^d$带权情形下建立了类似结果。

系数满足小Carleson条件的抛物算子的$L^p$ Neumann问题

The $L^p$ Neumann problem for parabolic operators with coefficients satisfying small Carleson condition

本文证明在系数满足小Carleson条件且区域Lipschitz常数足够小时，对任意$1<p<\infty$，Lipschitz柱面上的抛物Neumann问题可解。

具有BMO反对称部分的二阶椭圆算子的微扰理论

Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part

本文研究带无界BMO反对称部分的散度型椭圆算子的$L^p$狄利克雷问题微扰理论，证明系数差满足Carleson条件时可解性的保持与阶数关系，并将其应用于有界Lipschitz域的相关问题。

VMO泊松核消失Carleson条件必要性的反例

A Counterexample to the Necessity of the Vanishing Carleson Condition for VMO Poisson Kernels

通过双Lipschitz映射构造反例，证明消失Carleson条件不是VMO泊松核的必要条件，否定Kenig提出的问题。

Damek-Ricci 空间上具有径向初值的薛定谔算子解的 regularity 与逐点收敛性

Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces

本文研究 Damek-Ricci 空间上径向初值薛定谔方程的 Carleson 问题，证明了解逐点收敛的最优正则性指标为 $\alpha \ge 1/4$，与欧几里得情形一致。

椭圆算子在弦弧域上的扰动

Perturbations of elliptic operators in chord arc domains

研究了散度型算子解的边界正则性，这些算子是已知解的边界正则性的算子的小扰动。如果系数的偏差函数满足小范数Carleson测度条件，则一个算子是另一个算子的小扰动。本文将Escauriaza关于Lipschitz域的结果推广到具有小常数的弦弧域.特别地，我们证明了如果$L_1$是$L_0$的小扰动，并且$\log k_0$有小的BMO范数，那么$\log k_1$也有小的BMO范数.这里，$k_i$表示$L_i$的椭圆测度相对于域边界的表面测度的密度。

二阶复系数椭圆算子解的正则性理论与$L^p$狄利克雷问题

Regularity theory for solutions to second order elliptic operators with complex coefficients and the $L^p$ Dirichlet problem

当矩阵$A$的系数满足一个自然代数条件时，我们建立了一个新的二阶椭圆型复值方程$\mathcal L=$ div $A(\nabla\cdot)$的正则性理论，该条件是文献中已知的$L^p$ -耗散性条件的加强形式.准确地说，正则性结果是内球上解的$L^p$平均的逆Hölder条件，并作为实值散度型椭圆算子解的De Giorgi - Nash - Moser正则性的替代。Cialdea和Maz'ya在一系列论文中研究了二阶复系数算子和系统的$L^p$ -耗散性的充要条件。最近，Carbonaro和Dragičević引入了一个他们称之为$p$ -椭圆性的条件，并证明了它对某些由复值二阶微分算子产生的双线性算子的有界性有影响。他们的$p$ -椭圆性条件正是我们的$L^p$ -耗散性的加强版本.当$A$和$B$满足一个自然而熟悉的Carleson测度条件时，本文的正则性结果被应用于求解$\mathcal L=$ div $A(\nabla\cdot)+B\cdot\nabla$的$L^p$ Dirichlet问题.证明了当$A$是$p$ -椭圆时，$p$的$L^p$ Dirichlet边值问题的可解性.

Klein-Gordon流的逐点收敛性

Pointwise convergence of the Klein-Gordon flow

研究三次非线性Klein-Gordon方程的Carleson问题，证明Sobolev空间和随机化初始数据下解逐点收敛到初值的最优正则性。

Regularity for the two phase singular perturbation problems

We prove that an a priori BMO gradient estimate for the two phase singular perturbation problem implies Lipschitz regularity for the limits. This problem arises in the mathematical theory of combustion where the reaction-diffusion is modelled by the $p$-Laplacian. A key tool in our approach is the weak energy identity. Our method proves a natural and intrinsic characterization of the free boundary points and can be applied to more general classes of solutions.

Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Let $Ω\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), i.e., a set which satisfies the interior Corkscrew and Harnack chain conditions, respectively scale-invariant/quantitative versions of openness and path-connectedness. Assume that $Ω$ satisfies the so-called capacity density condition. Let $L_0u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$ be two real (non-necessarily symmetric) uniformly elliptic operators, and write $ω_{L_0}$, $ω_L$ for the associated elliptic measures. The goal of this program is to find sufficient c…

Regularity of fractional heat semigroup associated with Schrödinger operators

Let $L=-Δ+V$ be a Schrödinger operator, where the potential $V$ belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup $\{e^{-t{L}^α}\}_{t>0}, α>0$, associated with ${L}$. By the aid of the fundamental solution of the heat equation: $$\partial_{t}u+L u=\partial_{t}u -Δu+Vu=0,$$ we estimate the gradient and the time-fractional derivatives of the fractional heat kernel $K^{L}_{α,t}(\cdot, \cdot)$, respectively. This method is independent of the Fourier transform, and can be applied to the second order differential operators whose heat kernel…

波映射的整体正则性Ⅴ.能量类中的大数据局部适定性和扰动理论

Global regularity of wave maps V. Large data local wellposedness and perturbation theory in the energy class

利用调和映射热流和Tataru及作者的函数空间，建立了二维Minkowski空间$\R^{1+2}$到双曲空间$\H^m$的波映射在能量类上的一个大数据局部适定性结果.这是本系列早期文章中证明这种波映射的全局正则性所需的五个主张之一。

BMO空间中热方程柯西问题的全局时间极大正则性及其应用

Global-in-time maximal regularity for the Cauchy problem of the heat equation in BMO and applications

本文在BMO空间中建立了热方程柯西问题的全局时间极大正则性，并将方法推广到具有反向Hölder类位势的薛定谔算子，核心贡献在于利用热核估计、时空分解和平均振荡技巧获得全局衰减阶数。

上半空间中椭圆型方程组的狄利克雷问题及VMO的定量刻画

The BMO-Dirichlet problem for elliptic systems in the upper-half space and quantitative characterizations of VMO

本文证明了对于任何齐次二阶常复系数椭圆型方程组$L$，边界数据为BMO的$L$中的狄利克雷问题在函数类$u$中适定，其中$dμ_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$为Carleson测度.本文建立了一个Fatou型定理，保证了满足Carleson测度条件的系统的光滑零解$u$的点态非切边界迹的存在性.这意味着BMO可以被刻画为光滑零解$u$到椭圆系统$L$的非切逐点迹的集合，并且具有$μ_u$是Carleson测度的性质.我们建立了上半空间中BMO-狄利克雷问题的一个正则性结果：满足上述Carleson测度条件的$L$的任意给定光滑零解的非切点态迹属于Sarason空间VMO当且仅当$μ_u$满足一个消失Carleson测度条件.此外，我们得到的Dirichlet问题的适定性时，边界数据规定在Morrey-Campanato和解决方案需要满足消失Carleson测量条件的分数阶。因此，我们将空间VMO刻画为BMO中包含的光滑函数类的闭包，其中一致连续性可以适当量化（例如满足Hölder或Lipschitz条件的光滑函数类），改进了Sarason的经典结果，将VMO描述为BMO中具有有界平均振荡的一致连续函数空间的闭包。最后，证明了满足$T(1)=0$的Calderón-Zygmund算子$T$在$\mathrm{VMO}$中扩张为有界线性算子…

Interior epsilon-regularity theory for the solutions of the magneto-micropolar equations with a perturbation term

We develop here a particular version of the partial regularity theory for the Magneto-Micropolar equations (MMP) where a perturbation term is added. These equations are used in some special cases, such as in the study of the evolution of liquid cristals or polymers, where the classical Navier-Stokes equations are not an accurate enough model. The incompressible Magneto-Micropolar system is composed of three coupled equations: the first one is based in the Navier-Stokes system, the second one considers mainly the magnetic field while the last equation introduces the microrotation field represen…

具有横向独立系数的抛物算子的 $L^p$ 正则性问题

The $L^p$ regularity problem for parabolic operators with transversally independent coefficients

本文研究区域R^{n+1}_+×R上具有横向独立系数的抛物算子L^p正则性问题，证明存在p_0>1使得问题在(1,p_0)范围内可解，补充了Dirichlet问题的已有结果。

Boundary value problems for second order elliptic operators with complex coefficients

The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions $u$ to $\mathcal{L}u = 0$ will satisfy a higher integrability, even though they may not be continuous in the interior. Moreover, these solutions have the property that $|u|^{p/2-1}u \in W^{1,2}_{loc}$. These properties of solutions were used by Dindoš-Pipher to solve the $L^p$ Dirichlet problem for $p$-elliptic operators whose coefficients satisfy a further re…

具有径向正则性的短程波速扰动下的对数波衰减

Logarithmic wave decay for short range wavespeed perturbations with radial regularity

该论文针对二维及以上空间中具有短程径向正则波速扰动的波动方程，建立了局部能量对数衰减，方法上通过低频Neumann展开与高频Carleman估计分别处理预解式。

Maximal $L^2$-regularity in nonlinear gradient systems and perturbations of sublinear growth

The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function $φ$ has a smoothing effect, discovered by H. Brézis, which implies maximal regularity for the evolution equation. We use this and Schaefer's fixed point theorem to solve the evolution equation perturbed by a Nemytskii-operator of sublinear growth. For this, we need that the sublevel sets of $φ$ are not only closed but even compact. We apply our results to the $p$-Laplacian and also to the Dirichlet-to-Neumann operator with respect to $p$-harmonic functions.

抛物型泊松核的一个自由边界问题

A Free Boundary Problem for the Parabolic Poisson Kernel

研究了Hofmann，Lewis和Nyström提出的抛物弦弧域，证明了在连续阈值以下的一个自由边界正则性结果.更精确地说，我们表明，一个Reifenberg平坦，抛物线弦弧域的泊松核有对数VMO实际上必须是一个消失的弦弧域（即满足消失Carleson测量条件）。这概括了，抛物线设置，凯尼格和托罗的结果和答案肯定的一个问题，在上述文件中的霍夫曼等人。在这个证明的一个关键步骤是一个分类的“平坦”爆破的抛物线问题。

Counterexamples for high-degree generalizations of the Schrödinger maximal operator

In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space $H^s(\mathbb{R}^n)$ that implies pointwise convergence for the solution of the linear Schrödinger equation. After progress by many authors, this was recently resolved (up to the endpoint) by Bourgain, whose counterexample construction for the Schrödinger maximal operator proved a necessary condition on the regularity, and Du and Zhang, who proved a sufficient condition. Analogues of Carleson's question remain open for many other dispersive PDE's. We develop a flexible new method to approa…

Damek-Ricci 空间上具渐近凹相位的色散方程的正则性与逐点收敛

Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces

研究 Damek-Ricci 空间上具渐近凹相位的色散方程的 Carleson 问题，获得解向径向初始数据几乎处处逐点收敛的几乎尖锐正则性阈值 $β>a/4$，该结果推广了经典欧氏结论。

The existence and regularity theory for abstract semilinear time-fractional evolution equations

In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We will extend well-known results for standard evolution equations such as the blow-up alternative, to the time-fractional evolution equations. We also prove the differentiability with respect to time of the solution when the perturbation is sufficiently smooth. The differentiability enables us to use the maximum principle. The theory on general Banach spaces en…

非光滑椭圆型方程组II的加权极大正则估计和可解性

Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II

我们继续发展，通过减少到一阶系统的余法梯度，$L^2$ \textit{a priori}估计和解的边值问题的狄利克雷，正则性，纽曼型散度型二阶，复杂的椭圆系统。我们在这里工作的单位球和更一般的双Lipschitz图像，假设一个Carleson条件介绍了Dahlberg的措施的差异系数，他们的边界跟踪附近的边界。我们锐化我们的估计证明了一个一般的结果，关于\textit{a priori}几乎处处非切向收敛的边界。此外，紧的边界产生更多的可解性结果使用Fredholm理论。类之间的解决方案和唯一性问题的比较进行了讨论。因此，我们能够解决一个长期存在的正则性问题，真正的方程，这可能不是真的上半空间，证明\textit{a posteriori}一个单独的工作有界域。

单边弦弧域上椭圆算子的扰动。第一部分：对称算子的小扰动和大扰动

Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators

设$Ω\subset\mathbb{R}^{n+1}$，$n\ge 2$是一个单侧弦弧域，即满足内部Corkscrew和Harnack链条件（它们分别是开性和路连通性的尺度不变/定量形式）的域，其边界$\partialΩ$是$n$维Ahlfors正则的.考虑$L_0$和$L$两个实对称散度型椭圆算子，并设$ω_{L_0}$，$ω_L$为相应的椭圆测度.证明了如果$ω_{L_0}\in A_\infty(σ)$，其中$σ=H^n|_{\partialΩ}$，和$L$是$L_0$的扰动（在这个意义上，$L_0$和$L$之间的差异满足一定的Carleson测度条件），则$ω_L\in A_\infty(σ)$ .此外，如果$L$是$L_0$的一个足够小的扰动，那么我们可以保留逆Hölder类，也就是说，如果对于某些$1<p<\infty$，我们有$ω_{L_0}\in RH_p(σ)$，那么就有$ω_{L}\in RH_p(σ)$。等价地，如果$L^{p'}(σ)$中的Dirichlet问题对于$L_0$是可解的，那么对于$L$也是如此。这些结果可以看作是Dahlberg，February man-Kenig-Pipher和Milakis-Pipher-Toro在更良性的环境中得到的扰动定理的推广。作为我们的方法的结果，我们可以证明，对于拉普拉斯算子的任何扰动（或者更一般地说，任何具有满…

分数Poisson问题局部化finite differences的定量估计及其在正则性和谱稳定性中的应用

Quantitative estimates on localized finite differences for the fractional Poisson problem, and applications to regularity and spectral stability

我们建立了新的定量估计的局部有限差分的解决方案的泊松问题的分数Laplace算子齐次Dirichlet条件的固体型解决在有界区域满足Lipschitz锥正则性条件。然后我们应用这些估计得到（i）Besov空间中分数Poisson问题解的~正则性结果;（ii）分数Poisson问题解关于区域扰动的~定量稳定性估计;（iii）分数Laplace算子特征值和特征函数关于区域扰动的~定量稳定性估计.

非自治左、右乘扰动与极大正则性

Non-autonomous right and left multiplicative perturbations and maximal regularity

本文考虑非自治柯西问题$u'(t) + B(t)A(t)u(t) + P(t)u(t) = f(t), u(0) = u_0$和$u'(t) + A(t)B(t)u(t) + P(t)u(t) = f (t), u(0) = u_0$的极大正则性问题。在这两种情况下，不定常的运营商$A(t)$与一个家庭的半双线性型和乘法左或右扰动$B(t)$以及添加剂扰动$P(t)$是家庭的有界算子上考虑的希尔伯特空间。我们证明了最大的$L_p$ -正则性结果和其他正则性质的解决方案的最小正则性假设下的形式和扰动。

p（x）-Laplacian非齐次自由边界问题的弱解与界面正则性

Weak Solutions and Regularity of the Interface in an Inhomogeneous Free Boundary Problem for the p(x)-Laplacian

本文研究了p（x）-Laplacian的一个单相自由边界问题，其右端非零.证明了弱解的自由边界是每个自由边界点邻域中的C^1曲面。我们还得到了进一步的正则性结果的自由边界，进一步的正则性假设的数据。我们将这些结果应用于p（x）-Laplacian的非齐次奇异扰动问题的极限函数，我们在Lederman，C.，& Wolanski，N.本文研究了p（x）-Laplacian非线性微分方程的非齐次奇异扰动问题。138（2016），300-325.

演化方程中临界奇异扰动的最大正则性

Maximal regularity for evolution equations with critical singular perturbations

本文研究了演化方程中由临界奇异扰动引起的最大正则性问题，提出了一种加权最大正则性理论，扩展了之前的研究成果。

二阶非散度型算子的L^p狄利克雷问题：可解性和扰动结果

The L^p Dirichlet problem for second-order, non-divergence form operators: solvability and perturbation results

建立了非散度型算子L = A \nabla^2的Dahlberg扰动定理.如果L_0和L_1是Lipschitz域上的两个算子，使得L_0的L^p狄利克雷问题对p是可解的（1，\infty），并且两个算子的系数在Carleson测度意义下足够接近，则算子L_p的L^p狄利克雷问题这是Rios在“The L^p Diriclet problem and nondivergence调和测度“（Trans.AMS 355，2（2003））中证明的该结果的A_{\infty}版本的改进。因此，我们还改进了Dindos、Petermichl和Pipher的结果，“二阶椭圆算子和p适应平方函数的L^p狄利克雷问题”（J. Fun. Anal. 249（2007））通过大大削弱对算子系数所需的条件来证明非散度型算子的L^p可解性。改进后的条件与散度型算子L = div A \nabla的条件完全相同。

Global regularity of optimal mappings in non-convex domains

In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of optimal transportation in convex domains by the authors.

非局部方程的正则性结果

Regularity results for nonlocal equations by approximation

本文证明了Cordes-Niremberg估计的一个非局部形式。我们用它来扩展我们以前的正则性结果完全非线性积分微分方程的变系数的情况下和其他几个设置，方程可以被认为是一个扰动的平移不变的情况下。

波映射VII的全局正则性。离域或分散溶液的控制

Global regularity of wave maps VII. Control of delocalised or dispersed solutions

这是系列\cite{tao:heatwave}，\cite{tao:heatwave2}，\cite{tao:heatwave3}，\cite{tao:heatwave4}的最后一篇论文，它建立了二维波映射到双曲目标的全局正则性。在本文中，我们建立了这一声明所需的其余权利要求，即一个可分割的扰动理论，和一种手段的频率离域，空间分散，或空间离域数据的解决方案，严格较小的能量合成的解决方案。作为这里的扰动理论和本系列前面得到的结果的结果，我们还建立了波映射到双曲空间的时空边界和散射性质。

A priori estimates for the derivative nonlinear Schrödinger equation

We prove low regularity a priori estimates for the derivative nonlinear Schrödinger equation in Besov spaces with positive regularity index conditional upon small $L^2$-norm. This covers the full subcritical range. We use the power series expansion of the perturbation determinant introduced by Killip--Vişan--Zhang for completely integrable PDE. This makes it possible to derive low regularity conservation laws from the perturbation determinant.

The simultaneous asymmetric perturbation method for overdetermined free boundary problems

In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions that would have been out of reach otherwise. Another benefit of this method lies in the improvement of the regularity gap between the free boundary and the given one. Finally, some g…

Regularity for evolution equations with non-autonomous perturbations in Banach spaces

We provide regularity of solutions to a large class of evolution equations on Banach spaces where the generator is composed of a static principal part plus a non-autonomous perturbation. Regularity is examined with respect to the graph norm of the iterations of the principal part. The results are applied to the Schrödinger equation and conditions on a time-dependent scalar potential for regularity of the solution in higher Sobolev spaces are derived.

具有两个小参数的奇异摄动反应-对流-扩散边值问题的解析正则性

Analytic regularity for a singularly perturbed reaction-convection-diffusion boundary value problem with two small parameters

本文研究含两个小参数的奇异摄动反应-对流-扩散边值问题，建立了显式依赖于微分阶数和摄动参数的正则性界，并证明解可分解为光滑部分、端点边界层和可忽略余项。

全局亚椭圆伪微分算子的扰动

Perturbations of globally hypoelliptic pseudo-differential operators on $\mathbb{R}^n$

本文研究了全局亚椭圆伪微分算子在低阶扰动下的稳定性，证明其全局正则性得以保持，并探讨了其在标准Hörmander类中的不适用性。

Scattering problems from slightly perturbed periodic surfaces: Part I. regularity of the Bloch transformed fields

Rough surface scattering problems are always very challenging both theoretically and numerically. In this paper, we adopt the Bloch transform and the perturbation theory to investigate a special case, i.e., when the rough surface is a slight perturbation of a periodic one. Based on known results from the Bloch transform, the problem could be written into an equivalent bounded variational problem in higher dimensional spaces. The first step is to consider the non-perturbed problem. From the perturbation theory, the solution could be written as a Neumann series. The second step is to consider th…

具有表面张力的二维水波的整体正则性

Global regularity for 2d water waves with surface tension

我们认为，充分无旋水波系统的表面张力和没有重力在二维（毛细波系统），并证明了全球的正则性和修改散射适当小和局部扰动的平坦界面。我们的分析的一个重要的一点是开发一个足够强大的方法，能量估计和色散分析的基础上，这使我们能够处理强奇异性所产生的正常形式的方法和非线性散射的应用程序中的时间共振。因此，我们能够考虑一个合适的一类具有有限能量的扰动，但没有其他的动量条件。我们的分析的一部分依赖于一个新的治疗的狄利克雷-诺依曼算子在二维这是独立的利益。因此，本文的结果是完备的。

On the admissibility of observation operators in the context of maximal regularity

We study admissible observation operators for perturbed evolution equations using the concept of maximal regularity. We first show the invariance of the maximal $L^p$-regularity under non-autonomous Miyadera-Voigt perturbations. Second, we establish the invariance of admissibility of observation operators under such a class of perturbations. Finally, we illustrate our result with two examples, one on a non-autonomous parabolic system, and the other on an evolution equation with mixed boundary conditions and a non-local perturbation.

边界$\varepsilon$ -最优运输中的正则性

Boundary $\varepsilon$-regularity in optimal transportation

我们发展了一个$\e$ -正则性理论在边界上的一般类Monge-Ampère型方程中产生的最优运输。作为推论，我们推导出$C^2$一致凸区域上支持的Hölder密度之间的最优输运映射是$C^{1,α}$直到边界，只要费用函数是二次代价$-x\cdot y$的足够小的扰动。

高维周期薛定谔极大算子的尖锐下界

Sharp lower bounds for the periodic maximal Schrödinger operator in higher dimensions

本文证明了高维环面上薛定谔极大函数的尖锐下界，结合已有充分性结果，完全解决了周期薛定谔方程Carleson问题，并反驳了Miao-Yuan-Zhao猜想，方法基于阿贝尔簇上的复乘法。

关于Korteweg-de Vries方程某些扰动解的衰减性研究

On decay of solutions to some perturbations of the Korteweg-de Vries equation

该研究聚焦加权及非对称加权Sobolev空间中Korteweg-de Vries方程耗散扰动解的正则性与衰减性关系，旨在揭示二者关联。