Cristiana De Filippis, Giuseppe Mingione
Local Schauder estimates hold in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic variational problems and elliptic equations are locally Hölder continuous, provided coefficients are locally Hölder continuous.
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The sharp growth rate in nonuniformly elliptic Schauder theory
该研究探讨非一致椭圆Schauder理论，得出在椭圆率增长的最优假设下，Schauder估计对非一致椭圆问题成立的结论。
Sketches of Nonuniformly Elliptic Schauder Theory
本文概述了 Schauder 理论在非一致椭圆框架下的近期进展，特别是解决了非线性、非一致椭圆情形下 Schauder 估计有效性的长期难题。
We establish sharp global regularity results for solutions to nonhomogeneous, nonunifomrly elliptic systems with zero boundary conditions. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the forcing term, thus positively settling the optimality issue raised in \cite{bdms}.
We introduce and study a new functional which was motivated by our paper on the Caffarelli-Kohn-Nirenberg inequality with variable exponent (Bahrouni, Rădulescu and Repovš, Nonlinearity 31 (2018), 1518-1534). We also study the eigenvalue problem for equations involving this new functional.
Schauder-type estimates for fully nonlinear degenerate elliptic equations
该文针对完全非线性退化椭圆方程，研究其解在内部源项消失点处的正则性，得到了依赖类Hölder源项-椭圆性消失速率的Schauder型正则性估计。
We consider the pointwise in space Lp-type regularity for elliptic and parabolic equations of order m in Rn. We provide pointwise Schauder estimates for the general range of Lp exponents, extending previous results from p > n/m to 1 < p < n/m.
Lipschitz bounds for nonuniformly elliptic integral functionals in the plane
该研究针对二维平面带受控$(p,q)$增长的标量积分泛函局部极小元的正则性问题，将Lipschitz连续性成立的条件从经典的$q<2p$放宽至$q<3p$，同时得到了平面线性一致椭圆方程解的最优$L^\infty$-$L^2$估计。
We establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving Hölder semi-norms not with respect to all, but only with respect to some of the independent variables.
The volume potential for elliptic differential operators in Schauder spaces
本文证明了负指数Schauder空间中常系数二阶微分算子基本解对应的体积位势的连续性结果，并推广了正指数Schauder空间中的部分经典结果。
Under various conditions, we establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving Hölder semi-norms not with respect to all, but only with respect to some of the independent variables. A novelty of our results is that the coefficients are allowed to be merely measurable with respect to the other independent variables.
Parabolic and Elliptic Schauder Theory on Manifolds for a Fourth-Order Problem with a First- and a Third-Order Boundary Condition
本文介绍流形上的 Schauder 理论，推导了光滑紧致流形上具一阶和三阶边界条件的四阶抛物线性问题的 Schauder 估计，并借此证明了解的存在性且研究了相关椭圆问题。
Giaquinta's example revisited: the Liouville property for a class of nonuniformly elliptic equations in low dimensions
本文针对含凸密度的非一致椭圆方程，在低维条件下研究其整体解的Liouville性质，明确了有界解必为常数的判定条件。
Regularity for nonuniformly elliptic equations with $p,q-$growth and explicit $x,u-$dependence
本文研究散度型椭圆方程在 $p,q-$增长及显式 $(x,u)$ 依赖下弱解的局部有界性与 Lipschitz 正则性，并给出统一假设及处理新困难的方法。
Schauder estimates for a class of fully nonlinear elliptic PDEs: a geometric tangential approach
本文通过爆破论证、几何切向分析与紧致性技术，为非凸非凹完全非线性椭圆PDE的经典解建立了局部与全局Schauder估计。
We prove $m$-dimensional symmetry results, that we call $m$-Liouville theorems, for stable and monotone solutions of the following nonuniformly elliptic equation \begin{eqnarray*}\label{mainequ} - div(γ(\mathbf x') \nabla u(\mathbf x)) =λ(\mathbf x' ) f(u(\mathbf x)) \ \ \text{for}\ \ \mathbf x=(\mathbf x',\mathbf x'')\in\mathbf{R}^d\times\mathbf{R}^{s}=\mathbf{R}^n, \end{eqnarray*} where $0\le m<n$ and $0<λ,γ$ are smooth functions and $f\in C^1(\mathbf R)$. The interesting fact is that the decay assumptions on the weight function $γ(\mathbf x') $ play the fundamental role in deriving $m$-Liou…
Schauder and Cordes-Nirenberg estimates for nonlocal elliptic equations with singular kernels
该研究针对带奇异核的变系数非局部椭圆方程，证明了散度与非散度形式下的一般性Schauder型估计，还建立了系数小振荡条件下无$x$正则性假设的椭圆方程Hölder估计，首次涵盖一类新型算子。
We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+σ}$ and either Dini or Hölder continuous data. Here $0 < σ< 2$ is a constant and $a$ is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz--Zygmund spaces $Λ^{α+σ}$ and $Λ^α$ for any $α>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.
Schauder--Orlicz-Type Estimates for Divergence-Form Elliptic Equations with Lower-Order Terms
针对带低阶项的二阶线性散度型椭圆方程的弱解，推导了Schauder-Orlicz型估计，识别了低阶数据的最优关联空间，并推广至重排不变Banach函数空间。
Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients
本文针对退化或奇异椭圆方程，在DMO系数条件下建立边界Schauder型估计，并应用得到高阶边界Harnack原理。
Continuity of the double layer potential of a second order elliptic differential operator in Schauder spaces on the boundary
本文在指数大于等于二的Schauder空间中，证明了常系数非齐次二阶椭圆微分算子基本解对应的双层位势在边界上具有正则化性质，从而改进并推广了多位学者的经典结果及Kirsch关于Helmholtz算子的工作。
We consider a class of possibly degenerate second order elliptic operators $\cal A$ on $\R^n$. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving ${\cal A}$. The Hölder function spaces are defined with respect to a non-euclidean metric related to the operator $\cal A$.
Notes on Schauder estimates by scaling for elliptic PDEs in divergence form
本文通过缩放方法，系统介绍了椭圆偏微分方程在散度形式下的Schauder估计，结合几何技巧和刚性结果，为自由边界问题研究提供理论基础。
Schauder and Calderón-Zygmund type estimates for fully nonlinear parabolic equations under "small ellipticity aperture" and applications
本文针对小椭圆孔径下的非凸完全非线性二阶抛物方程，推导了粘性解的Schauder估计、有界数据下的C^{1, Log-Lip}光滑性及Calderón-Zygmund估计，并将结果与相关最新估计建立关联。
One method available to prove the Schauder estimates is Neil Trudinger's method of mollification. In the case of second order elliptic equations, the method requires little more than mollification and the solid mean value inequality for subharmonic functions. Our goal in this article is show how the mean value property of subsolutions of the heat equation can be used in a similar fashion as the solid mean value inequality for subharmonic functions in Trudinger's original elliptic treatment, providing a relatively simple derivation of the interior Schauder estimate for second order parabolic eq…
We study elliptic equations on bounded domain of Euclidean spaces in the variable Hölder spaces. Interior a priori Schauder estimates are given as well as global ones. Moreover, the existence and the uniqueness of solutions to the Dirichlet boundary value problem is proved.
The present paper aims to generalize the Schauder estimate for a class of higher-order hypo-elliptic operators. The results in the present paper apply to parabolic equations of higher order and, for example, operators like $\partial_t^2-\partial_x^3$. Generalization to systems is also directly obtained.
Nonuniform ellipticity in variational problems and regularity
本文概述了非均匀椭圆问题正则性理论的近期进展，重点关注变分性质的问题。
In this paper, we establish pointwise Schauder estimates for solutions of nonlocal fully nonlinear elliptic equations by perturbative arguments. A key ingredient is a recursive Evans-Krylov theorem for nonlocal fully nonlinear translation invariant equations.
Global Schauder Regularity and Convergence for Uniformly Degenerate Parabolic Equations
该文研究一致退化抛物方程解的全局Hölder正则性，并在极限一致退化椭圆方程特征指数的附加假设下，分析时间趋于无穷时解的收敛性。
Schauder estimates for flat solutions to a class of fully nonlinear elliptic PDEs with Dini continuous data: a geometric tangential approach
本文通过几何切线方法，建立了具有Dini连续数据的全非线性椭圆PDE的局部Schauder估计，并扩展了相关理论。
We study the local regularity of solutions $f$ to the integro-differential equation $$ Af=g \quad \text{in $U$}$$ associated with the infinitesimal generator $A$ of a Lévy process $(X_t)_{t \geq 0}$. Under the assumption that the transition density of $(X_t)_{t \geq 0}$ satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions $f$. Our results apply for a wide class of Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions.
We consider a second-order parabolic equation in $\bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the $L_{\infty}$-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case o…
Schauder type estimates for degenerate or singular parabolic systems with partially DMO coefficients
本文针对退化或奇异系数的椭圆和抛物系统，在部分Dini平均振荡条件下建立边界Schauder型估计，并将高阶边界Harnack原理推广至任意阶。
We establish interior Schauder estimates for kinetic equations with integro-differential diffusion. We study equations of the form $f_t + v \cdot \nabla_x f = \mathcal L_v f + c$, where $\mathcal L_v$ is an integro-differential diffusion operator of order $2s$ acting in the $v$-variable. Under suitable ellipticity and Hölder continuity conditions on the kernel of $\mathcal L_v$, we obtain an a priori estimate for $f$ in a properly scaled Hölder space.
Interior Schauder estimates for fractional elliptic equations in nondivergence form
本文研究非散度型分数阶椭圆算子的非局部 Poisson 问题，通过退化/奇异延拓、新型粘性解及 Caffarelli 扰动方法建立精确内部 Schauder 估计，并证明 Hopf 引理、Harnack 不等式与 Hölder 正则性。
A duality approach to gradient Hölder estimates for linear divergence form elliptic equations
本文提出了一种基于对偶性的方法，用于推导线性散度形式椭圆方程的梯度Hölder估计，通过Schauder理论与局部Hardy空间和Hölder空间的联系，展示了稀疏界和反Hölder不等式的推导方法。
We establish global Schauder estimates for integro-partial differential equations (IPDE) driven by a possibly degenerate Lévy Ornstein-Uhlenbeck operator, both in the elliptic and parabolic setting, using some suitable anisotropic Hölder spaces. The class of operators we consider is composed by a linear drift plus a Lévy operator that is comparable, in a suitable sense, with a possibly truncated stable operator. It includes for example, the relativistic, the tempered, the layered or the Lamperti stable operators. Our method does not assume neither the symmetry of the Lévy operator nor the inva…
Schauder estimates for elliptic equations degenerating on lower dimensional manifolds
本文研究在低维特征流形上退化或奇异的椭圆方程，通过穿孔、爆破和新 Liouville 定理建立弱解直至退化流形的 Schauder 型正则性。
We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerate-elliptic operators of the kind described in our article appear in a diverse range of applications, including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance, generators of diffusion processes arising in mathematical b…
This article sets forth results on the existence, a priori estimates and boundedness of positive solutions of a singular quasilinear systems of elliptic equations involving variable exponents. The approach is based on Schauder's fixed point Theorem. A Moser iteration procedure is also obtained for singular cooperative systems involving variable exponents establishing a priori estimates and boundedness of solutions.
A survey on the boundary behavior of the double layer potential in Schauder spaces in the frame of an abstract approach
本文综述了非齐次二阶椭圆算子双层位势边界积分算子在Hölder和Schauder空间中的连续性结果，并展示这些性质可由抽象度量空间框架推导。
Generalized Evans--Krylov and Schauder type estimates for nonlocal fully nonlinear equations with rough kernels of variable orders
该研究针对带变阶粗糙核的非局部完全非线性椭圆方程，引入描述可变可微阶数的函数与广义Hölder空间，建立了推广标准结果的广义Evans--Krylov型与Schauder型$C^{\phi\psi}$估计。
Schauder-type Estimates and Log-Critical Well-posedness for the Two-Phase Muskat Problem with Surface Tension
针对具有表面张力、不同黏度和密度对比的两相Muskat问题，通过推导适应对数临界尺度的Schauder型估计，证明了在任意维度下大初始数据的短时间适定性。
A note on relativistic kinetic Schauder estimates
该文针对带相对论输运的线性动理学方程，构造仅基于动量的Schauder估计的反例，证明完整Lorentzian Hölder控制无法被仅基于动量的Hölder控制替代。
The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains
本文针对低维边界上具有奇异系数的加权椭圆方程Dirichlet问题，结合穿孔域逼近、Liouville型定理和爆破论证，建立了直到边界的最优Schauder正则性估计。
Schauder-type estimates and applications
本文系统阐述 Schauder 型估计在椭圆型偏微分方程理论中的核心地位，并给出实际应用中最常用定理的完整证明，揭示求解 PDE 与获得先验界之间的等价关系。
Perturbative Schauder Estimates for $Q$-Valued Quasilinear Elliptic Systems and a Sharp Dimension Bound for Branch Sets of Stationary Graphs
本文为$Q$-值拟线性椭圆系统建立微扰Schauder估计，并证明平稳图的分支集维数不超过$n-2$，该界在余维数一时是锐利的。
In this paper, we derive an interior Schauder estimate for the divergence form elliptic equation \begin{equation*} D_i(a(x)D_iu)=D_if_i \end{equation*} in $\mathbb{R}^2$, where $a(x)$ and $f_i(x)$ are piecewise Hölder continuous in a domain containing two touching balls as subdomains. When $f_i\equiv 0$ and $a$ is piecewise constant, we prove that $u$ is piecewise smooth with bounded derivatives. This completely answers a question raised by Li and Vogelius [7] in dimension 2.
Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to understand Schauder type estimates fashioning particular borderline scenarios. In such context, it has been a common accepted aphorism that the continuity of the Hessian of a solution could never be superior than the continuity of the medium. Notwithstanding, in this article we show that solutions to uniformly elliptic, linear equations with $C^{0,\barε}$ coefficien…
High-Order regularity for fully nonlinear elliptic transmission problems under weak convexity assumption
本文研究了由二阶完全非线性一致椭圆方程建模的传输问题的Schauder理论，针对非凹非凸算子F，在小的椭圆孔径和超水平集凸性两种情形下证明了高阶正则性结果。
For nonautonomous, nonuniformly elliptic integrals with so-called $(p,q)$-growth conditions, we show a general interpolation property allowing to get basic higher integrability results for Hölder continuous minimizers under improved bounds for the gap $q/p$. For this we introduce a new method, based on approximating the original, local functional, with mixed local/nonlocal ones, and allowing for suitable estimates in fractional Sobolev spaces.
Mixed type boundary value problem of elliptic equation in a thin domain
本文研究薄新月形区域中带斜导数与狄利克雷混合边界条件的二维椭圆方程，证明了区域塌缩为线段时边值问题具有一致 Schauder 估计并给出渐近估计。
Schauder estimates for equations with cone metrics, II
本文研究具锥奇点方程的线性理论，针对沿简单正规交叉除子具锥奇点背景 Kähler 度量的线性椭圆与抛物方程导出内部 Schauder 估计，并以此证明 conical Kähler-Ricci flow 的短时存在性。
We study local boundedness for subsolutions of nonlinear nonuniformly elliptic equations whose prototype is given by $\nabla \cdot (λ|\nabla u|^{p-2}\nabla u)=0$, where the variable coefficient $0\leqλ$ and its inverse $λ^{-1}$ are allowed to be unbounded. Assuming certain integrability conditions on $λ$ and $λ^{-1}$ depending on $p$ and the dimension, we show local boundedness. Moreover, we provide counterexamples to regularity showing that the integrability conditions are optimal for every $p>1$.
Higher Hölder regularity for degenerate fully nonlinear elliptic equations with Hamiltonian terms
针对带变指数双相退化律和哈密顿项的完全非线性椭圆方程，在哈密顿系数和源项消失的点处获得梯度Hölder正则性改进，并在局部极值处建立关于消失率的Sharp Schauder型估计。
We establish the well-posedness of a coupled micro-macro parabolic-elliptic system modeling the interplay between two pressures in a gas-liquid mixture close to equilibrium that is filling a porous media with distributed microstructures. Additionally, we prove a local stability estimate for the inverse micro-macro Robin problem, potentially useful in identifying quantitatively a micro-macro interfacial Robin transfer coefficient given microscopic measurements on accessible fixed interfaces. To tackle the solvability issue we use two-scale energy estimates and two-scale regularity/compactness a…
A priori regularity estimates for equations degenerating on nodal sets
本文研究了退化椭圆方程在节点集上的正则性估计，通过Schauder估计和放大论证，证明了更高阶边界Harnack原理。
This paper studies the Sobolev regularity of weak solution of degenerate elliptic equations in divergence form $\text{div}[\mathbf{A}(X) \nabla u] = \text{div}[\mathbf{F}(X)]$, where $X = (x,y) \in \mathbb{R}^{n} \times \mathbb{R}$ . The coefficient matrix $\mathbf{A}(X)$ is a symmetric, measurable $(n+1) \times (n+1)$ matrix, and it could be degenerate or singular in the one dimensional $y$-variable as a weight function in the Muckenhoupt class $A_2$ of weights. Our results give weighted Sobolev regularity estimates of Calderón-Zygmund type for weak solutions of this class of singular, degene…
A priori estimates for solutions of degenerate fully nonlinear elliptic equations with $L^p$ data
本文研究了具有L^p数据的退化全非线性椭圆方程的粘性解的先验正则性估计，证明了当非齐次项属于L^p(p>n)时的最优内部C^{1,α}估计，在临界情况下，在Lorentz条件f∈L^{n,1}下获得对数Lipschitz连续性模，利用滑动抛物面或尖点方法开发了在合适梯度 regime 中的均匀Hölder估计，并通过修正论方法建立了一个近似引理，从而推导出相应的Schauder型估计。
We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new relations between these two equations. In particular, we show that solutions to an autonomous Beltrami equation enjoy a quantitative improved degree of Hölder regularity, higher than what is given by the classical exponent $1/K$.
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