 NSFPAR ID:
 10427419
 Date Published:
 Journal Name:
 Advances in Calculus of Variations
 Volume:
 0
 Issue:
 0
 ISSN:
 18648258
 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 (aka 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 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 (nonnecessarily 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 chordarc 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 upperhalf space, or nontangentially 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, chordarc, or 1sided chordarc domains, our methods allow us to immediately recover a number of existing perturbation results as well as extend some of them.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 $ 
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

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