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 Hölder classes, that is, if for some $1
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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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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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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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Let \(\Sigma\) be a closed subset of \(\mathbb{R}^{n+1}\) which is parabolic Ahlfors-David regular and assume that \(\Sigma\) satisfies a 2-sided corkscrew condition. Assume, in addition, that \(\Sigma\) is either time-forwards Ahlfors-David regular, time-backwards Ahlfors-David regular, or parabolic uniform rectifiable. We then first prove that \(\Sigma\) satisfies a weak synchronized two cube condition. Based on this we are able to revisit the argument of Nyström and Strömqvist (2009) and prove that \(\Sigma\) contain suniform big pieces of Lip(1,1/2) graphs. When \(\Sigma\) is parabolic uniformly rectifiable the construction can be refined and in this case we prove that \(\Sigma\) contains uniform big pieces of regular parabolic Lip(1,1/2) graphs. Similar results hold if \(\Omega\subset\mathbb{R}^{n+1}\) is a connected component of \(\mathbb{R}^{n+1}\setminus\Sigma\) and in this context we also give a parabolic counterpart of the main result of Azzam et al. (2017) by proving that if \(\Omega\) is a one-sided parabolic chord arc domain, and if \(\Sigma\) is parabolic uniformly rectifiable, then \(\Omega\) is in fact a parabolic chord arc domain. Our results give a flexible parabolic version of the classical (elliptic) result of David and Jerison (1990) concerning the existence of uniform big pieces of Lipschitz graphs for sets satisfying a two disc condition.
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