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1. Optimal Poisson kernel regularity for elliptic operators with H ̈older-continuous coefficients in vanishing chord-arc domains

   Bortz, Simon; Toro, Tatiana; Zhao, Zihue (September 2023, Journal of functional analysis)

   We show that if $$\Omega$$ is a vanishing chord-arc domain and $$L$$ is a divergence-form elliptic operator with H\"older-continuous coefficient matrix, then $$\log k_L \in VMO$, where $$k_L$$ is the elliptic kernel for $$L$$ in the domain $$\Omega$$. This extends the previous work of Kenig and Toro in the case of the Laplacian 

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

   https://doi.org/http://dx.doi.org/10.1090/tran/7536  

   Cavero, J.; Hofmann, S.; Martell, J.M. (January 2019, Transactions of the American Mathematical Society)

   Let $$\Omega\subset\re^{n+1}$$, $$n\ge 2$$, 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 $$\partial\Omega$$ is $$n$$-dimensional Ahlfors regular. Consider $$L_0$$ and $$L$$ two real symmetric divergence form elliptic operators and let $$\omega_{L_0}$$, $$\omega_L$$ be the associated elliptic measures. We show that if $$\omega_{L_0}\in A_\infty(\sigma)$$, where $$\sigma=H^n\lfloor_{\partial\Omega}$$, and $$L$$ is a perturbation of $$L_0$$ (in the sense that the discrepancy between $$L_0$$ and $$L$$ satisfies certain Carleson measure condition), then $$\omega_L\in A_\infty(\sigma)$$. Moreover, if $$L$$ is a sufficiently small perturbation of $$L_0$$, then one can preserve the reverse H√∂lder classes, that is, if for some $$1<\infty$$, one has $$\omega_{L_0}\in RH_p(\sigma)$$ then $$\omega_{L}\in RH_p(\sigma)$$. Equivalently, if the Dirichlet problem with data in $$L^{p'}(\sigma)$$ is solvable for $$L_0$$ then so it is for $$L$$. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg, Fefferman-Kenig-Pipher, and Milakis-Pipher-Toro in more benign settings. As a consequence of our methods we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $$A_\infty(\sigma)$$ then necessarily $$\Omega$$ is in fact an NTA domain (and hence chord-arc) and therefore its boundary is uniformly rectifiable. 

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3. Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

   https://doi.org/https://doi.org/10.1090/tran/6927  

   Akman, Murat; Badger, Matthew; Hofmann, Steve; Martell, Jose Maria (July 2017, Transactions of the American Mathematical Society)

   Let $$ \Omega \subset \mathbb{R}^{n+1}$$, $$ n\geq 2$$, be a 1-sided NTA domain (also known as a uniform domain), i.e., a domain which satisfies interior corkscrew and Harnack chain conditions, and assume that $$ \partial \Omega $$ is $ n$-dimensional Ahlfors-David regular. We characterize the rectifiability of $$ \partial \Omega $$ 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 $$ \partial \Omega $$ can be covered $$ \mathcal {H}^n$$-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of $$ \Omega $$ and to the fact that $$ \partial \Omega $$ possesses exterior corkscrew points in a qualitative way $$ \mathcal {H}^n$$-a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition. 

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4. Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

   https://doi.org/10.1515/acv-2021-0053  

   Akman, Murat; Hofmann, Steve; Martell, José María; Toro, Tatiana (May 2022, Advances in Calculus of Variations)

   Abstract Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1-sided non-tangentially accessible domain (also known as uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider two real-valued (non-necessarily symmetric) uniformly elliptic operators L 0 ⁢ u = - div ⁡ ( A 0 ⁢ ∇ ⁡ u )   and   L ⁢ u = - div ⁡ ( A ⁢ ∇ ⁡ u ) L_{0}u=-\operatorname{div}(A_{0}\nabla u)\quad\text{and}\quad Lu=-%\operatorname{div}(A\nabla u) in Ω, and write ω L 0 {\omega_{L_{0}}} and ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this article and its companion[M. Akman, S. Hofmann, J. M. Martell and T. Toro,Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition,preprint 2021, https://arxiv.org/abs/1901.08261v3 ]is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\operatorname{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper, we are interested in obtaininga square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work ofDahlberg, Jerison and Kenig and are fundamental for the proof of the perturbation results in the paper cited above. 

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5. Superconvergence of High Order Finite Difference Schemes Based on Variational Formulation for Elliptic Equations

   https://doi.org/https://doi.org/10.1007/s10915-020-01144-w  

   Li, Hao; Zhang, Xiangxiong (January 2020, Journal of scientific computing)

   The classical continuous finite element method with Lagrangian Q^k basis reduces to a finite difference scheme when all the integrals are replaced by the (𝑘+1)×(𝑘+1) Gauss–Lobatto quadrature. We prove that this finite difference scheme is (𝑘+2)-th order accurate in the discrete 2-norm for an elliptic equation with Dirichlet boundary conditions, which is a superconvergence result of function values. We also give a convenient implementation for the case 𝑘=2, which is a simple fourth order accurate elliptic solver on a rectangular domain. 

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