E. Milakis, T. Toro
We study the boundary regularity of solutions of elliptic operators in divergence form with $C^{0,α}$ coefficients or operators which are small perturbations of the Laplacian in non-smooth domains. We show that, as in the case of the Laplacian, there exists a close relationship between the regularity of the corresponding elliptic measure and the geometry of the domain.
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We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique $W^{1,p}$ solvability is obtained with $p$ being in the optimal range $(4/3,4)$. The leading coefficients are assumed to have small mean oscillations and the boundary of domain is Reifenberg flat. We also assume that the two boundary conditions are separated by some Reifenberg flat set of co-dimension $2$ on the boundary.
We provide a detailed proof of the fact that any domain which is sufficiently flat in the sense of Reifenberg is also Jones-flat, and hence it is an extension domain. We discuss various applications of this property, in particular we obtain L^\infty estimates for the eigenfunctions of the Laplace operator with Neumann boundary conditions. We also compare different ways of measuring the "distance" between two sufficiently close Reifenberg-flat domains. These results are pivotal to the quantitative stability analysis of the spectrum of the Neumann Laplacian performed in another paper by the same…
This paper is devoted to the investigation of the boundary regularity for the Poisson equation $${{cc} -Δu = f & \text{in} Ω u= 0 & \text{on} \partial Ω$$ where $f$ belongs to some $L^p(Ω)$ and $Ω$ is a Reifenberg-flat domain of $\mathbb R^n.$ More precisely, we prove that given an exponent $α\in (0,1)$, there exists an $\varepsilon>0$ such that the solution $u$ to the previous system is locally Hölder continuous provided that $Ω$ is $(\varepsilon,r_0)$-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem.
Hölder regularity up to the boundary for the g-Laplacian on Reifenberg flat domains
针对Reifenberg平坦域上具有非标准增长的非线性椭圆Dirichlet问题，通过结合迭代边界衰减论证和Orlicz-Sobolev框架下的ABP型极大值原理，证明了弱解在边界附近的α-Hölder连续性。
Nonlinear nonlocal equations in Reifenberg flat domains
本文研究 Reifenberg 平坦区域上非齐次分数阶 p-Laplace 方程，在超出 Lipschitz 范畴的非光滑有界域中建立了解及其梯度的精细边界正则性，且结果在线性情形下亦属首次。
Boundary regularity for nonlocal elliptic equations over Reifenberg flat domains
本文证明了Reifenberg平坦域上非局部椭圆方程解的尖锐边界正则性，即当平坦度参数足够小时解具有$C^{s-\varepsilon}$正则性，证明基于障碍函数和比较原理的归纳论证。
In this paper, we investigate the boundary Hölder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the p-Laplace equations and fully nonlinear elliptic equations) on Reifenberg flat domains. We prove that for any $0<α<1$, there exists $δ>0$ such that the solution is $C^α$ at $x_0\in \partial Ω$ provided that $Ω$ is $δ$-Reifenberg flat at $x_0$ (see Definition 1.1). In particular, for any $0 < α< 1$, if $\partial Ω$ is $C^1$ and $u=g$ on $\partial Ω$ with $g\in C^α(x_0)$, then $u\in C^α(x_0)$. A similar result for t…
We prove weighted $L_{p,q}$-estimates for divergence type higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains. In particular, in the parabolic case the coefficients do not have any regularity assumptions in the time variable. As functions of the spatial variables, the leading coefficients are permitted to have small mean oscillations. The weights are in the class of Muckenhoupt weights $A_p$. We also prove the solvability of the systems in weighted Sobolev spaces.
We study the obstacle problem with an elliptic operator in divergence form. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary in the case where the coefficients are in VMO.
Boundary Hölder regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle
本文研究Reifenberg平坦域上分数阶拉普拉斯方程的弱解，在平坦度参数足够小时，利用迭代论证和非局部ABP极大值原理，证明了弱解具有边界Hölder正则性。
Let P be an hyperplane in R^N, and denote by dH the Hausdorff distance. We show that for all positive radius r < 1 there is an epsilon > 0, such that if K is a Reifenberg-flat set in B(0; 1), a ball in R^N, that contains the origin, with d_H(K; P) <epsilon, and if u is an energy minimizing function in B(0; 1)\K with restricted values on @B(0; 1)\K, then the normalized energy of u in B(0; r)\K is bounded by the normalized energy of u in B(0; 1)\K. We also prove the same result in R^3 when K is a epsilon-minimal set, that is a generalization of Reifenberg-flat sets with minimal cones of type Y a…
A weighted norm inequality of Muckenhoupt-Wheeden type is obtained for gradients of solutions to a class of quasilinear equations with measure data on Reifenberg flat domains. This essentially leads to a resolution of an existence problem for quasilinear Riccati type equations with a gradient source term of arbitrary power law growth.
In this paper, we prove global weighted Lorentz and Lorentz-Morrey estimates for gradients of solutions to the quasilinear parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=\operatorname{div}(F),$$ in a bounded domain $Ω\times (0,T)\subset\mathbb{R}^{N+1}$, under minimal regularity assumptions on the boundary of domain and on nonlinearity $A$. Then results yields existence of a solution to the Riccati type parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=|\nabla u|^q+\operatorname{div}(F)+μ,$$ where $q>1$ and $μ$ is a bounded Radon measure.
In this paper, we study the set of absolute continuity of p-harmonic measure, $μ$, and $(n-1)-$dimensional Hausdorff measure, $\mathcal{H}^{n-1}$, on locally flat domains in $\mathbb{R}^{n}$, $n\geq 2$. We prove that for fixed $p$ with $2<p<\infty$ there exists a Reifenberg flat domain $Ω\subset\mathbb{R}^{n}$, $n\geq 2$, with $\mathcal{H}^{n-1}(\partialΩ)<\infty$ and a Borel set $K\subset\partialΩ$ such that $μ(K)>0=\mathcal{H}^{n-1}(K)$ where $μ$ is the p-harmonic measure associated to a positive weak solution to p-Laplace equation in $Ω$ with continuous boundary value zero on $\partialΩ$. W…
$W^{1, p}$ estimate for the solutions of elliptic equations whose coefficient matrix can have large jump along the boundary of subdomains is obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO seminorms. The domain and subdomains are Reifenberg flat domains and moreover, it has been shown that the estimates are uniform with respect to the distance between the subdomains.
Boundary Harnack inequalities for the fractional $p$-Laplacian in Reifenberg flat domains
该研究针对Reifenberg平坦区域，证明了分数阶p-调和函数商的边界哈纳克不等式与赫尔德连续性，通过系列估计和变形方法将相关结果从能量类扩展到尾类。
The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains
该研究针对小常数平面Reifenberg平坦区域上实值一致椭圆Lipschitz系数散度型算子的椭圆测度，证明其Hausdorff维数不超过1，且存在满椭圆测度的σ-有限一维Hausdorff测度边界子集，推广了Wolff关于调和测度的结果。
We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincaré inequality on Reifenberg flat domains, the proof of which is of independent interest.
We consider the mixed Dirichlet-conormal problem on irregular domains in $\mathbb{R}^d$. Two types of regularity results will be discussed: the $W^{1,p}$ regularity and a non-tangential maximal function estimate. The domain is assumed to be Reifenberg-flat, and the interfacial boundary is either Reifenberg-flat of co-dimension $2$ or is locally sufficiently close to a Lipschitz function of $m$ variables, where $m=1,\ldots,d-2$. For the non-tangential maximal function estimate, we also require the domain to be Lipschitz.
We prove the unique solvability of solutions in Sobolev spaces to the stationary Stokes system on a bounded Reifenberg flat domain when the coefficients are partially BMO functions, i.e., locally they are merely measurable in one direction and have small mean oscillations in the other directions. Using this result, we establish the unique solvability in Muckenhoupt type weighted Sobolev spaces for the system with partially BMO coefficients on a Reifenberg flat domain. We also present weighted a priori $L_q$-estimates for the system when the domain is the whole Euclidean space or a half space.
In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,\text{Dini}} condition at a boundary point, and the nonhomogeneous term satisfies Dini continuous condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg C^{1,\text{Dini}} domains.
In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type \begin{align*} \mathrm{div}(A(x,\nabla u)) = \mathrm{div}(f) \ \text{in} \ Ω, \mbox{ and } \ u = g \ \text{on} \ \partial Ω, \end{align*} that related to Riesz potentials. Here, in our setting, $Ω\subset \mathbb{R}^n$, $n \ge 2$ is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded me…
Let $Ω^+\subset\mathbb R^{n+1}$ be an NTA domain and let $Ω^-= \mathbb R^{n+1}\setminus \overline{Ω^+}$ be an NTA domain as well. Denote by $ω^+$ and $ω^-$ their respective harmonic measures. Assume that $Ω^+$ is a $δ$-Reifenberg flat domain for some $δ>0$ small enough. In this paper we show that $\log\frac{dω^-}{dω^+}\in VMO(ω^+)$ if and only if $Ω^+$ is vanishing Reifenberg flat, $Ω^+$ and $Ω^-$ have joint big pieces of chord-arc subdomains, and the inner unit normal of $Ω^+$ has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counte…
In this note, we prove the boundary Hölder regularity for the infinity Laplace equation under a proper geometric condition. This geometric condition is quite general, and the exterior cone condition, the Reifenberg flat domains, and the corkscrew domains (including the non-tangentially accessible domains) are special cases. The key idea, following [3], is that the strong maximum principle and the scaling invariance imply the boundary Hölder regularity.
We consider the mixed Dirichlet-conormal problem for the heat equation on cylindrical domains with a bounded and Lipschitz base $Ω\subset \mathbb{R}^d$ and a time-dependent separation $Λ$. Under certain mild regularity assumptions on $Λ$, we show that for any $q>1$ sufficiently close to 1, the mixed problem in $L_q$ is solvable. In other words, for any given Dirichlet data in the parabolic Riesz potential space $\mathcal{L}_q^1$ and the Neumann data in $L_q$, there is a unique solution and the non-tangential maximal function of its gradient is in $L_q$ on the lateral boundary of the domain. Wh…
The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.
This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data: \begin{cases} -\operatorname{div}(A(x,D u))=g-\operatorname{div} f \quad & \mathrm{in} \quad Ω\\ u= 0 \quad & \text{on} \ \partial Ω, \end{cases} where $Ω\subset \mathbb{R}^n$ is sufficiently flat in the sense of Reifenberg.
We obtain the global weighted $W^{1,p}$ estimates for weak solutions of nonlinear elliptic equations over Reifenberg flat domains. Where nonlinearity $A(x,z,ξ)$ is assumed to be local uniform continuous in $z$ and have small BMO semi-norm in $x$. Moreover, we derive Besov regularity for solutions of a class of special harmonic equations by making use of $W^{1,p}$ estimate. Keywords: global weighted $W^{1,p}$ estimates; quasilinear equations; Besov regularity
We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nyström, and prove a free boundary regularity result below the continuous threshold. More precisely, we show that a Reifenberg flat, parabolic chord arc domain whose Poisson kernel has logarithm in VMO must in fact be a vanishing chord arc domain (i.e. satisfies a vanishing Carleson measure condition). This generalizes, to the parabolic setting, a result of Kenig and Toro and answers in the affirmative a question left open in the aforementioned paper of Hofmann et al. A key step in this proof is a classification of "flat" b…
Boundary regularity for the polyharmonic Dirichlet problem
本研究针对Reifenberg平坦区域上的m重调和Poisson方程齐次Dirichlet问题，通过推广Nirenberg平移法并结合Mosco收敛工具，证明了解直到边界的$\mathscr{C}^{m-1,α}$正则性。
A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each $d>1$, there exists a Reifenberg flat domain $Ω\subset \mathbb{R}^{d+1}$ with $\mathcal{H}^{d}(\partialΩ)<\infty$ and a subset $E\subset \partial Ω$ with positive harmonic measure yet zero $\mathcal{H}^{d}$-measure. In particular, this implies that a classical theorem of F. and M. Riesz theorem fails in higher dimens…
We study global regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in rough domains $Ω$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $Ω$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.
A non-homogeneous mixed local and nonlocal problem in divergence form is investigated for the validity of the global Calderón-Zygmund estimate for the weak solution to the Dirichlet problem of a nonlinear elliptic equation. We establish an optimal Calderón-Zygmund theory by finding not only a minimal regularity requirement on the mixed local and nonlocal operators but also a lower level of geometric assumption on the boundary of the domain for the global gradient estimate. More precisely, assuming that the nonlinearity of the local operator, whose prototype is the classical $(-Δ_p)^1$-Laplace …
Global gradient estimates in weighted Lebesgue spaces for parabolic operators
本文研究了抛物算子在加权Lebesgue空间中的正则性问题，通过Calderón-Zygmund型理论得出弱解空间梯度的正则性结果。
In this paper we establish new quantitative stability estimates with respect to domain perturbations for all the eigenvalues of both the Neumann and the Dirichlet Laplacian. Our main results follow from an abstract lemma stating that it is actually sufficient to provide an estimate on suitable projection operators. Whereas this lemma could be applied under different regularity assumptions on the domain, here we use it to estimate the spectrum in Lipschitz and in so-called Reifenberg-flat domains. Our argument also relies on suitable extension techniques and on an estimate on the decay of the e…
We consider elliptic equations and systems in divergence form with the conormal or the Robin boundary conditions, with small BMO (bounded mean oscillation) or variably partially small BMO coefficients. We propose a new class of domains which are locally close to a half space (or convex domains) with respect to the Lebesgue measure in the system (or scalar, respectively) case, and obtain the $W^1_p$ estimate for the conormal problem with the homogeneous boundary condition. Such condition is weaker than the Reifenberg flatness condition, for which the closeness is measured in terms of the Hausdo…
We obtain the regularity of solutions in Sobolev spaces for the mixed Dirichlet-conormal problem for parabolic operators in cylindrical domains with time-dependent separations, which is the first of its kind. Assuming the boundary of the domain to be Reifenberg-flat and the separation to be locally sufficiently close to a Lipschitz function of $m$ variables, where $m=0,\ldots,d-2$, with respect to the Hausdorff distance, we prove the unique solvability for $p\in (2(m+2/(m+3),2(m+2)/(m+1)))$. In the case when $m=0$, the range $p\in(4/3,4)$ is optimal in view of the known results for Laplace equ…
We derive global gradient estimates for $W^{1,p}_0(Ω)$-weak solutions to quasilinear elliptic equations of the form $$ \mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over $n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to $x$ and Hölder continuous in $u.$ In the case when $p\geq n,$ we allow only continuous nonlinearity in $u.$ Our result highly improves the known regularity results available in the literature. In fact, we are able not only to weaken the regularity requirement on the nonl…
We obtain a global weighted $L^p$ estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, while the boundary of the domain is supposed to be Reifenberg flat, which goes beyond the category of domains with Lipschitz continuous boundaries. As consequence of the main result, we derive global gradient estimate for the weak solution in the framework of the Morrey spaces which implie…
We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the $p$-Laplacian type and in non-divergence form. We provide local Hölder and Lipschitz estimates for the solutions. In the degenerate case, we prove the Hölder regularity of the gradient. Our study is based on a combination of the method of alternatives and the improvement of flatness estimates.
We study a class of two-phase inhomogeneous free boundary problems governed by elliptic equations in divergence form. In particular we prove that Lipschitz or flat free boundaries are $C^{1,γ}$. Our results apply to the classical Prandtl-Bachelor model in fluiddynamics.
We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called $\varepsilon$-scale flatness condition, which could be arbitrarily rough below $\varepsilon$-scale. This particularly generalizes Kenig and Prange's work in [32] and [33] by a quantitative approach.
Boundary estimates for non-divergence equations in $C^1$ domains
本文在$C^1$领域中研究非散度方程的边界估计，通过显式连续模扩展了经典霍普-奥列尼克引理，并统一了利普希茨正则性理论。
We prove low frequency estimates for the boundary values of the resolvent of long range perturbations of the flat Laplacian in divergence form.
Solvability of divergence equation in Lipschitz spaces
研究有界\(C^2\)区域上齐次狄利克雷边界条件下散度方程\(\operatorname{div} \u = f\)的可解性，通过结合局部化技术与边界扁平化过程构造解\(\u\)，使其满足一定条件，给出了该方程可解性的相关结论。
Let $n\ge2$ and $Ω\subset\mathbb{R}^n$ be a bounded NTA domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second order elliptic equations of divergence form with an elliptic symmetric part and a BMO anti-symmetric part in $Ω$. More precisely, for any given $p\in(2,\infty)$, the authors prove that a weak reverse Hölder inequality with exponent $p$ implies the global $W^{1,p}$ estimate and the global weighted $W^{1,q}$ estimate, with $q\in[2,p]$ and some Muckenhoupt weights, of solutions to Dirichlet boundary value prob…
This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution in a low regularity set-up. We first discuss the case of zero obstacle and $W^{1,p}$ metrics with $p\in(n+1,\infty]$. In this framework, we prove the $C^{1,α}$ regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal $C^{1,\min\{1-\frac{n+1}{p}, \frac{1…
The two-phase problem for harmonic measure in VMO and the chord-arc condition
本文研究了VMO与弦弧条件下的调和测度两相问题，证明了Radon-Nikodym导数的对数属于VMO等价于区域为弦弧域且内法向属于VMO。
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