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1. Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition

   https://doi.org/10.1515/forum-2022-0323  

   Akman, Murat; Hofmann, Steve; Martell, José María; Toro, Tatiana (December 2022, Forum Mathematicum)

   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 (aka 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 L 0 ⁢ u = - div ⁢ ( A 0 ⁢ ∇ ⁡ u ) {L_{0}u=-\mathrm{div}(A_{0}\nabla u)} , L ⁢ u = - div ⁢ ( A ⁢ ∇ ⁡ u ) {Lu=-\mathrm{div}(A\nabla u)} , two real (non-necessarily symmetric) uniformly elliptic operators in Ω, and write ω L 0 {\omega_{L_{0}}} , ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\mathrm{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper we establish that if the discrepancy of the two matrices satisfies a natural Carleson measure condition with respect to ω L 0 {\omega_{L_{0}}} , then ω L ∈ A ∞ ⁢ ( ω L 0 ) {\omega_{L}\in A_{\infty}(\omega_{L_{0}})} . Additionally, we can prove that ω L ∈ RH q ⁢ ( ω L 0 ) {\omega_{L}\in\mathrm{RH}_{q}(\omega_{L_{0}})} for some specific q ∈ ( 1 , ∞ ) {q\in(1,\infty)} , by assuming that such Carleson condition holds with a sufficiently small constant. This “small constant” case extends previous work of Fefferman–Kenig–Pipher and Milakis–Pipher together with the last author of the present paper who considered symmetric operators in Lipschitz and bounded chord-arc domains, respectively. Here we go beyond those settings, our domains satisfy a capacity density condition which is much weaker than the existence of exterior Corkscrew balls. Moreover, their boundaries need not be Ahlfors regular and the restriction of the n -dimensional Hausdorff measure to the boundary could be even locally infinite. The “large constant” case, that is, the one on which we just assume that the discrepancy of the two matrices satisfies a Carleson measure condition, is new even in the case of nice domains (such as the unit ball, the upper-half space, or non-tangentially accessible domains) and in the case of symmetric operators. We emphasize that our results hold in the absence of a nice surface measure: all the analysis is done with the underlying measure ω L 0 {\omega_{L_{0}}} , which behaves well in the scenarios we are considering. When particularized to the setting of Lipschitz, chord-arc, or 1-sided chord-arc domains, our methods allow us to immediately recover a number of existing perturbation results as well as extend some of them. 

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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. The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

   https://doi.org/10.1007/s00208-021-02219-1  

   Hofmann, Steve; Li, Linhan; Mayboroda, Svitlana; Pipher, Jill (July 2021, Mathematische Annalen)

   Grafakos, Loukas (Ed.)

   We establish the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti- symmetric part. In particular, the coefficients are not necessarily bounded. We prove that for some $$p\in (1,\infty)$$, the Dirichlet problem for the elliptic equation $$Lu= \dv A\nabla u=0$$ in the upper half-space $$\mathbb{R}^{n+1},\, n\geq 2,$$ is uniquely solvable when the boundary data is in $$L^p(\mathbb{R}^n,dx)$$, provided that the coefficients are independent of the vertical variable. This result is equivalent to saying that the elliptic measure associated to $$L$$ belongs to the $$A_\infty$$ class with respect to the Lebesgue measure $dx$, a quantitative version of absolute continuity. 

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4. 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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5. BMO solvability and absolute continuity of harmonic measure

   https://doi.org/http://dx.doi.org/10.1007/s12220-017-9959-0  

   Hofmann, S.; Le, P. (January 2018, The Journal of geometric analysis)

   We show that for a uniformly elliptic divergence form operator $$L$$, defined in an open set $$\Omega$$ with Ahlfors-David regular boundary, BMO-solvability implies scale invariant quantitative absolute continuity (the weak-$$A_\infty$$ property) of elliptic-harmonic measure with respect to surface measure on $$\partial \Omega$$. We do not impose any connectivity hypothesis, qualitative or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of $$\Omega$$. In this generality, our results are new even for the Laplacian. Moreover, we obtain a converse, under the additional assumption that $$\Omega$$ satisfies an interior Corkscrew condition, in the special case that $$L$$ is the Laplacian. 

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