E. Milakis, J. Pipher, T. Toro
In the present paper we study the solvability of the Dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients which are small perturbations of given operators in rough domains beyond the Lipschitz category. In our approach, the development of the theory of tent spaces on these domains is essential.
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We study the chord-arc Jordan curves that satisfy the Cotlar-type inequality $T_*(f)\lesssim M^2(Tf),$ where $T$ is the Cauchy transform, $T_*$ is the maximal Cauchy transform and $M$ is the Hardy-Littlewood maximal function. Under the background assumption of asymptotic quasi-conformality we find a characterization of such curves in terms of the smoothness of a parametrization of the curve.
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。
We study the boundary regularity of solutions to divergence form operators which are small perturbations of operators for which the boundary regularity of solutions is known. An operator is a small perturbation of another operator if the deviation function of the coefficients satisfies a Carleson measure condition with small norm. We extend Escauriaza's result on Lipschitz domains to chord arc domains with small constant. In particular we prove that if $L_1$ is a small perturbation of $L_0$ and $\log k_0$ has small BMO norm so does $\log k_1$. Here $k_i$ denotes the density of the elliptic mea…
Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains
本文针对单侧弦弧域中带形式有界漂移的椭圆算子，建立边值与位势理论，推导相关估计与性质，最终利用AI构造出具有维数下降调和测度的特定算子。
We show that if $Ω$ is a vanishing chord-arc domain and $L$ is a divergence-form elliptic operator with Hölder-continuous coefficient matrix, then $\log k_L \in VMO$, where $k_L$ is the elliptic kernel for $L$ in the domain $Ω$. This extends the previous work of Kenig and Toro in the case of the Laplacian.
Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains
本文研究了在弦域中椭圆算子的Neumann问题的可解性，证明了在满足特定条件时，该问题在L^p空间中可解。
We show that if $Ω\subset \mathbb{R}^{n+1}$, $n\geq 1$, is a uniform domain (aka 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of $Ω$ implies the existence of exterior Corkscrew points at all scales, so that in fact, $Ω$ is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior Corkscrew conditions, and an interior Harnack Chain condition. We discuss some implications of this result, for theorems of F. and M. Riesz type, and for certain f…
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…
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…
Let $Ω\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $\partialΩ$ is $n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric divergence form elliptic operators and let $ω_{L_0}$, $ω_L$ be the associated elliptic measures. We show that if $ω_{L_0}\in A_\infty(σ)$, where $σ=H^n|_{\partialΩ}$, and $L$ is a perturbation of $L_0$ (in the sense that the discrepa…
We study the properties of harmonic measure in semi-uniform domains. Aikawa and Hirata showed in \cite{AH08} that, for John domains satisfying the capacity density condition (CDC), the doubling property for harmonic measure is equivalent to the domain being semi-uniform. Our first result removes the John condition by showing that any domain satisfying the CDC whose harmonic measure is doubling is semi-uniform. Next, we develop a substitute for some classical estimates on harmonic measure in nontangentially accessible domains that works in semi-uniform domains. We also show that semi-uniform do…
In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $L^p$, for some finite $p$, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $A_\infty$. In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundar…
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…
Let $Ω\subset \mathbb{R}^{n+1}$ be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that $\partialΩ$ may be approximated in a "Big Pieces" sense by boundaries of chord-arc subdomains of $Ω$, and hence that harmonic measure for $Ω$ is weak-$A_\infty$ with respect to surface measure on $\partialΩ$, provided that $Ω$ satisfies a certain weak version of a local John condition. Under the further assumption that $Ω$ satisfies an interior Corkscrew condition, and combined with our previous work, and with recent work of Azzam, Mourgoglou and Tolsa…
Open Inextensible Filaments in Planar Stokes Flow: Well-Posedness, Endpoint Asymptotics, and Straightening
本文研究平面 Stokes 流中自由端不可伸展开放细丝的动力学，通过三阶非局部曲率方程证明近临界数据的局部适定性、端点渐近展开，并给出小扰动及有限能量下全局存在与指数拉直结果。
Boundary unique continuation in planar domains by conformal mapping
本文通过共形映射证明了平面域中非平凡调和函数在边界上连续消失的性质，改进了Kenig和Zhao的近期结果，研究了临界点的有限性。
According to the Schwarz symmetry principle, every harmonic function vanishing on a real analytic curve has an odd continuation, while a harmonic function satisfying homogeneous Neumann condition has the even continuation. There are different generalizations of the Schwarz symmetry principle. Most of them are dealing with homogeneous conditions and the Dirichlet case. Using a technique of Dirichlet to Neumann and Robin to Neumann operators, we derive reflection formulae for nonhomogeneous Neumann and Robin conditions from a reflection formula subject to a nonhomogeneous Dirichlet condition.
Connectivity conditions and boundary Poincaré inequalities
该研究探讨了余维1 Ahlfors–David正则边界开集的连通性条件与边界庞加莱不等式的关联，证明了局部John加外部开瓶器条件可推出Harnack链条件、双侧弦弧域边界支持Heinonen–Koskela型弱1-庞加莱不等式，并构造了反例。
A counterexample regarding a two-phase problem for harmonic measure in VMO
本文通过在 $\mathbb R^2$ 中构造反例，否定了消失 Reifenberg 平坦区域中调和测度对数导数对数属于 VMO 这一悬而未决问题，且该反例的外单位法向量在恒为常数。
Trace estimates for harmonic functions along circular arcs with applications to domain decomposition on overlapping disks
本文推导了调和函数沿圆弧的 $\mathrm{L}^2$-迹范数及 DtD 算子的几何相关估计，并将其应用于二维重叠圆盘的 Schwarz 区域分解方法，为理论化学 COSMO 模型的离散并行 Schwarz 方法建立了严格收敛证明与误差估计。
Let $Ω\subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that $\partialΩ$ is $n$-dimensional Ahlfors-David regular. We characterize the rectifiability of $\partialΩ$ in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that $\partialΩ$ can be covered $\mathcal{H}^n$-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of $Ω$ and to the fact that $\partialΩ$ p…
We show that if $0<t<s\leq n-1$, $Ω\subseteq \mathbb{R}^{n}$ with lower $s$-content regular complement, and $z\in Ω$, there is a chord-arc domain $Ω_{z}\subseteq Ω$ with center $z$ so that $\mathscr{H}^{t}_{\infty}(\partialΩ_{z}\cap \partialΩ)\gtrsim_{t} \textrm{dist}(z,Ω^{c})^{t}$. This was originally shown by Koskela, Nandi, and Nicolau with John domains in place of chord-arc domains when $n=2$, $s=1$, and $Ω$ is a simply connected planar domain. Domains satisfying the conclusion of this result support $(p,β)$-Hardy inequalities for $β<p-n+t$ by a result of Koskela and Lehrbäck; Lehrbäck als…
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…
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