Regularity theory for nonlinear integral operators
Luis Caffarelli, Chi Hin Chan, Alexis Vasseur
本文证明了一类非线性积分变分问题经典解的存在性。这些问题涉及非局部图像和信号处理。
This article is dedicated to the proof of the existence of classical solutions for a class of non-linear integral variational problems. Those problems are involved in nonlocal image and signal processing.
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This paper considers hyperbolic wave equations with non-local in time conditions involving integrals with respect to time. It is shown that regularity of the solution can be achieved for complexified problem with integral conditions involving harmonic complex exponential weights. The paper establishes existence, uniqueness, and a regularity of the solutions.
A singular integral approach to the maximal L^p regularity of parabolic equations
本文描述了一种简单而基本的方法来求抛物型方程的极大L^p正则性，它只使用Volterra型奇异积分的概念.不需要解析半群、R-有界性或H^ \infty -谓词演算的知识。
Partial regularity for variational integrals with Morrey-Hölder zero-order terms, and the limit exponent in Massari's regularity theorem
本文研究了变分积分中带有Morrey-Hölder零阶项的部分正则性，验证了Massari正则性定理中极限指数的最优正则性结果。
We revisit the question of regularity for minimizers of scalar autonomous integral functionals with so-called $(p,q)$-growth. In particular, we establish Lipschitz regularity under the condition $\frac{q}p<1+\frac{2}{n-1}$ for $n\geq3$ which improves a classical result due to Marcellini~[JDE'91].
Regularity for Minimizers of Mixed-Smoothness Variational Integrals
该研究针对各向异性Sobolev空间上的拟凸变分积分极小元，建立了首个部分正则性结果，相关积分用于 pantographic 超材料的高阶梯度连续介质模型。
We prove regularity properties in the vector valued case for minimizers of variational integrals of the form $$A(u) = \int_ΩA(x,u,Du) dx$$ where the integrand $A(x,u,Du)$ is not necessarily continuous respect to the variable $x,$ grows polinomially like $|ξ|^p,$ $p \geq 2.$
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。
Optimal regularity of Fourier integral operators with one-sided folds
当两个投影之一是Whitney折叠时，我们得到了与奇异标准关系相关的傅里叶积分算子在索伯列夫空间中的最优连续性.正则性取决于来自典范关系的另一个投影的类型$k$（$k=1$表示惠特尼折叠）。证明了导数在正则性中失去了$(4+\frac{2}{k})^{-1}$.证明是基于振荡积分算子的$L^2$估计。
We establish the local Lipschitz continuity and the higher differentiability of vector-valued local minimizers of a class of energy integrals of the Calculus of Variations. The main novelty is that we deal with possibly degenerate energy densities with respect to the x-variable.
In this paper, we prove boundary pointwise $C^{k,α}$ regularity for any $k\geq 1$ for fully nonlinear parabolic equations. As an application, we give a direct and short proof of the higher regularity of the free boundaries in obstacle-type problems.
Existence and regularity of solutions in nonlinear wave equations
本文研究一类描述非线性波的非线性偏微分方程的Hölder连续解的整体存在性和正则性.
Regularity of the solutions for nonlinear biharmonic equations in R^{N}
建立了一类非线性双调和方程弱解的正则性.
Regularity properties of nonlinear Schrödinger equations in vector-valued spaces
本文得到了向量值函数空间中线性和非线性抽象薛定谔方程柯西问题解的正则性.
In this paper, we prove a boundary pointwise regularity for fully nonlinear elliptic equations on cones. In addition, based on this regularity, we give simple proofs of the Liouville theorems on cones.
On the regularity of solutions to a class of nonlinear dispersive equations
我们将研究某些非线性色散模型解的特殊正则性。我们的目标是展示如何将初始数据的规律性转移到解决方案中。这将取决于测量规则性的空间。
$C^{1,α}$ Interior Regularity for Nonlinear Nonlocal Elliptic Equations With Rough Kernels
在不假设核的正则性的情况下，证明了完全非线性一致椭圆型积分微分方程的一个$C^{1,α}$内正则性定理.然后，我们给出了一些应用的线性理论和更高的正则性的一类特殊的非线性算子。
A Visual Approach to the Regularity Theory for Fully Nonlinear Elliptic Equations
本文通过视觉方法重新审视一致椭圆方程的正则性理论，旨在提供直观理解。
Hölder regularity of viscosity solutions of some fully nonlinear equations in the Heisenberg group
本文证明了Heisenberg群中一类退化的完全非线性方程的有界、一致连续、黏性解的Hölder正则性。
We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.
In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form \[ \mathfrak{F}( v, Ω)= \int_Ω \! F(x, Dv(x)) \, dx, \] where, for ${n > 2}$ and $N\ge 1$, $Ω$ is a bounded open set in $\mathbb{R}^n$, $u \in W^{1,1}(Ω, \mathbb{R}^N)$ and the energy density $F:Ω\times \mathbb{R}^{N \times n}\to \mathbb{R}$ satisfies the so called variable growth conditions. The main novelty of the paper is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the $x$ variable.
Regularity for Fully Nonlinear P-Laplacian Parabolic Systems: the Degenerate Case
研究一类强耦合退化抛物方程组有界弱解的Hölder正则性.
Regularity for minimizers of scalar integral functionals
本文在$(p,q)$-增长条件下，证明了标量积分泛函局部极小子的局部Lipschitz正则性，其核心贡献在于去除了对能量密度关于$ξ$变量特殊结构的假设。
Interior regularity for nonlocal fully nonlinear equations with Dini continuous terms
研究了具有Dini连续项的非平移不变非局部完全非线性方程黏性解的内部正则性。利用摄动方法和递归Evans-Krylov定理得到了非局部方程的$C^σ$正则性估计.
Regularity results for fully nonlinear parabolic integro-differential operators
本文考虑具有对称核的完全非线性抛物型积分微分方程的正则性理论。我们能够找到Alexandrov-Backelman-Pucci估计的抛物线版本，其中0<σ<2。通过获得解的水平集的衰减估计，我们证明了解的Harnack不等式、Hölder正则性和C^{1，α}-正则性。
We show how to infer sharp partial regularity results for relaxed minimizers of degenerate, nonuniformly elliptic quasiconvex functionals, using tools from Nonlinear Potential Theory. In particular, in the setting of functionals with $(p,q)$-growth - according to the terminology of Marcellini [52] - we derive optimal local regularity criteria under minimal assumptions on the data.
A regularity theorem for fully nonlinear maximally subelliptic PDE
本文研究完全非线性极大次椭圆 PDE 的内部正则性，分别在非各向同性、各向同性 Sobolev 空间及相关经典框架中建立精确定理。
We prove the partial Hölder continuity for minimizers of quasiconvex functionals \[ \mathcal{F}({\bf u}) \colon =\int_Ω f(x,{\bf u},D{\bf u})\,\mathrm{d}x, \] where $f$ satisfies a uniform VMO condition with respect to the $x$-variable and is continuous with respect to ${\bf u}$. The growth condition with respect to the gradient variable is assumed a general one.
We prove global Lipschitz regularity for a wide class of convex variational integrals among all functions in $W^{1,1}$ with prescribed (sufficiently regular) boundary values, which are not assumed to satisfy any geometrical constraint (as for example bounded slope condition). Furthermore, we do not assume any restrictive assumption on the geometry of the domain and the result is valid for all sufficiently smooth domains. The result is achieved with a suitable approximation of the functional together with a new construction of appropriate barrier functions.
This short note is the announcement of a forthcoming work in which we prove a first general boundary regularity result for area-minimizing currents in higher codimension, without any geometric assumption on the boundary, except that it is an embedded submanifold of a Riemannian manifold, with a mild amount of smoothness ($C^{3, a_0}$ for a positive $a_0$ suffices). Our theorem allows to answer a question posed by Almgren at the end of his Big Regularity Paper. In this note we discuss the ideas of the proof and we also announce a theorem which shows that the boundary regularity is in general we…
Hölder regularity of the gradient for solutions of fully nonlinear equations with sublinear first order terms
本文建立了一类一阶项为次线性项的完全非线性方程解的正则性结果。我们证明了局部Hölder正则性的梯度在内部和边界。
We obtain up to a flat boundary regularity results in parabolic Hölder spaces for viscosity solutions of fully nonlinear parabolic equations with oblique boundary conditions.
We prove that viscosity solutions to fully nonlinear elliptic equations with degeneracy of double phase type are locally $C^{1,γ}$-regular.
Interior $C^{2,α}$ regularity for fully nonlinear uniformly elliptic equations in dimension two
本文研究了二维全非线性均匀椭圆方程粘性解的内部$C^{2,α}$正则性问题。
Regularity of viscous solutions for a degenerate nonlinear Cauchy problem
考虑一类带强迫项的非线性退化抛物型方程的Cauchy问题.利用粘性消失法得到了该方程的广义解。我们证明了这类广义解的一些正则性结果。
We study the continuity in weighted Fourier Lebesgue spaces for a class of pseudodifferential operators, whose symbol has finite Fourier Lebesgue regularity with respect to $x$ and satisfies a quasi-homogeneous decay of derivatives with respect to the $ξ$ variable. Applications to Fourier Lebesgue microlocal regularity of linear and nonlinear partial differential equations are given.
Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity
本文考虑一类非线性薛定谔方程的Cauchy问题，该方程具有一个只依赖于$\bar{u}$的导数非线性项.在亚临界正则标度下，A。Grünrock（2000年）。利用Hadac，Herr和Koch（2009）证明KP-Ⅱ方程在尺度临界正则性下的适定性和解的散射性的$U^{2}$空间和$V^{2}$空间，证明了方程在尺度临界正则性下的适定性和解的散射性.
Gradient regularity for fully nonlinear equations with degenerate coefficients
针对在超曲面上退化的完全非线性方程，推导了粘性解的$C^{1,α}$正则性估计。
Lipschitz regularity results for nonlinear strictly elliptic equations and applications
非线性严格椭圆型方程的Lipschitz正则性大多是在非线性项关于梯度变量（例如次二次项）的适当增长幂下得到的。对于具有超二次梯度增长幂的方程，人们通常使用弱Bernstein型参数，这需要对梯度非线性进行正则性和/或凸型假设。本文利用Ishii-Lions方法中的一些思想，得到了一类Hamilton函数关于梯度变量具有任意增长幂的非线性严格椭圆型方程的Lipschitz正则性结果.我们利用这些界来解决一个遍历问题，并研究了发展方程解的正则性和大时间行为。
Global low regularity solutions for nonlinear elastic waves
研究了三维非线性弹性波在低正则性初值下满足零条件的Cauchy问题。在径向对称情形下，我们证明了低权$H^3\times H^2$中每个小数据点的低正则解的整体存在性。
Holder regularity for fully nonlinear nonlocal equations
本综述证明带局部梯度项的完全非线性非局部一致椭圆方程的粘性解具有Hölder正则性，推广了Barles等人2011年的结果至极值非局部算子。
Regularity for radial solutions of degenerate fully nonlinear equations
本文证明了当算子是Hessian算子、Hessian算子中的一次齐次算子、梯度算子中的某次齐次算子$α>-1$以及当梯度不为零时的椭圆算子时，完全非线性方程径向解的导数的Holder正则性.
Logarithmical regularity criteria in terms of pressure for the three dimensional nonlinear dissipative system modeling electro-diffusion
本文在齐次Besov空间中建立了Navier-Stokes/Poisson-Nernst-Planck系统的数学改进的正则性判据。
We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, α}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator.
Regularity for anisotropic fully nonlinear integro-differential equations
我们考虑完全非线性积分微分方程的内核，在不同的方向上有不同的齐次性。我们证明了ABP估计的非局部版本，Harnack不等式和内部$C^{1, γ}$正则性，将[Caffarelli and Silvestre，CPAM 62，2009]的结果扩展到各向异性的情况。
A counterexample to $C^{2,1}$ regularity for parabolic fully nonlinear equations
我们解决了奇异抛物问题的自相似可解性，并表明抛物型完全非线性方程的解决方案预计不会是$C^{2,1}$。
Regularity theory for fully nonlinear integro-differential equations
我们考虑非线性积分微分方程，如从随机控制问题产生的纯跳Lèvy过程。我们得到了一般完全非线性积分微分方程的ABP估计、Harnack不等式和内$C^{1,α}$正则性的非局部形式。我们的估计保持一致的程度的方程接近2，所以它们可以被看作是一个自然的椭圆偏微分方程的正则性理论的延伸。
In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise $C^α$, $C^{1,α}$ and $C^{2,α}$ regularity. As byproducts, we also prove the A-B-P maximum principle, Harnack inequality, uniqueness and solvability of the equations.
We prove the partial Hölder continuity on boundary points for minimizers of quasiconvex non-degenerate functionals \begin{equation*} \mathcal{F}({\bf u}) \colon =\int_Ω f(x,{\bf u},D{\bf u})\,\mathrm{d}x, \end{equation*} where $f$ satisfies a uniform VMO condition with respect to the $x$-variable, is continuous with respect to ${\bf u}$ and has a general growth with respect to the gradient variable.
Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker-Planck equations
本文证明了二维空间无边界耦合Navier-斯托克斯方程和Fokker-Planck方程解的整体正则性。证明产生空间梯度的增长的先验估计。
Regularity theory for fully nonlinear oblique transmission problems
针对完全非线性斜向传输问题，建立了平坦界面下粘性解的最优正则性，得到分段C^{1,α}正则性，并推广到变系数方程。
The interior free boundary theory for linear elliptic operators in higher dimensions was developed by Caffarelli in the low regularity context. In these notes, the up-to-the boundary free boundary regularity is discussed for nonlinear elliptic operators based on a different approach.
$L^p$-regularity of a geometrically nonlinear system in supercritical dimensions
本文研究了源自几何非线性平坦Cosserat微极模型的系统，在超临界维度下建立了内部正则性和精确的$L^p$正则性，扩展了Gastel和Neff在临界维度下的工作。
We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class $C_{loc}^{1, α}$, for some constant $α\in (0, 1)$. In addition, under suitable conditions on $σ$, we prove regularity estimates in Hölder spaces for any viscosity solution. We also examine the singular setting and prove Hölder regularity estimates for the gradient of the solutions.
Asymptotics and regularity of flat solutions to fully nonlinear elliptic problems
在本文中，我们建立了非凸完全非线性椭圆方程平坦解的局部$C^{2,α}$正则性估计，条件是系数和源函数都是$C^{0,α}$类。对于仅仅是连续数据的问题，我们证明了平坦解是局部的$C^{1,\text{Log-Lip}}$。
We establish an $\varepsilon$-regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth, and, as a consequence, the partial regularity of such BV minimizers. This result extends the regularity theory for minimizers of quasiconvex integrals on Sobolev spaces to the context of maps of bounded variation. Previous partial regularity results for BV minimizers in the linear growth set-up were confined to the convex situation.
We consider nonlinear Schr{ö}dinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. To do so, we recast the theory of multiphase weakly nonlinear geometric optics for nonlinear Schr{ö}dinger equations in a general abstract functional setting.
In this note, we prove $\mathcal{C}^{1,γ}$ regularity for solutions of some fully nonlinear degenerate elliptic equations with "superlinear" and "subquadratic " Hamiltonian terms. As an application, we complete the results of \cite{BDL1} concerning the associated ergodic problem, proving, among other facts, the uniqueness, up to constants, of the ergodic function.
Gradient regularity for degenerate fully nonlinear free transmission problems with Hamiltonian terms
研究含哈密顿项的退化完全非线性自由传输问题，通过置于粘性不等式背景建立梯度局部赫尔德正则性，基于新估计与局部分析获逐点\(C^{1,\alpha}\)正则性。
Regularity of viscosity solutions to fully nonlinear elliptic transmission problems
该研究建立完全非线性二阶一致椭圆方程传输问题粘性解的正则性理论，完善平界面传输问题的相关基础理论，并推导曲面情形下粘性解直至传输面的多阶Hölder正则性结果。
Well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation
证明了非线性四阶薛定谔方程（NL4 S）在索伯列夫空间中的局部适定性.我们还研究了次临界情形下解的正则性。这种正则性的一个直接结果是在某些假设下，在质量和能量空间上的整体适定性。最后，我们证明了（NL4S）在某些情况下的超临界范围的不适定性。
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