 NSFPAR ID:
 10427413
 Date Published:
 Journal Name:
 Forum Mathematicum
 Volume:
 35
 Issue:
 1
 ISSN:
 09337741
 Page Range / eLocation ID:
 245 to 295
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1sided nontangentially accessible domain (also known as uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scaleinvariant/quantitative versions of openness and pathconnectedness. Let us assume also that Ω satisfies the socalled capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider two realvalued (nonnecessarily 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 1sided 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 nontangential estimates for solutions of operators as before. We establish that bounded weak nullsolutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak nullsolution, the associated square function can be controlled by the nontangential 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.more » « less

Let $\Omega\subset\re^{n+1}$, $n\ge 2$, be a 1sided chordarc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scaleinvariant/quantitative versions of the openness and pathconnectedness), 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

Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
arXiv:2010.09793 ) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators associated to a domain$$L_{\beta ,\gamma } = {\text {div}}D^{d+1+\gamma n} \nabla $$ ${L}_{\beta ,\gamma}=\text{div}{D}^{d+1+\gamma n}\nabla $ with a uniformly rectifiable boundary$$\Omega \subset {\mathbb {R}}^n$$ $\Omega \subset {R}^{n}$ of dimension$$\Gamma $$ $\Gamma $ , the now usual distance to the boundary$$d < n1$$ $d<n1$ given by$$D = D_\beta $$ $D={D}_{\beta}$ for$$D_\beta (X)^{\beta } = \int _{\Gamma } Xy^{d\beta } d\sigma (y)$$ ${D}_{\beta}{\left(X\right)}^{\beta}={\int}_{\Gamma}{Xy}^{d\beta}d\sigma \left(y\right)$ , where$$X \in \Omega $$ $X\in \Omega $ and$$\beta >0$$ $\beta >0$ . In this paper we show that the Green function$$\gamma \in (1,1)$$ $\gamma \in (1,1)$G for , with pole at infinity, is well approximated by multiples of$$L_{\beta ,\gamma }$$ ${L}_{\beta ,\gamma}$ , in the sense that the function$$D^{1\gamma }$$ ${D}^{1\gamma}$ satisfies a Carleson measure estimate on$$\big  D\nabla \big (\ln \big ( \frac{G}{D^{1\gamma }} \big )\big )\big ^2$$ $D\nabla (ln(\frac{G}{{D}^{1\gamma}})){}^{2}$ . We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).$$\Omega $$ $\Omega $ 
Let $ \Omega \subset \mathbb{R}^{n+1}$, $ n\geq 2$, be a 1sided 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 AhlforsDavid 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 chordarc 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.more » « less

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 nonsmooth 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 halfspace $\mathbb{R}^{n+1},\, n\geq 2,$ 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.more » « less