Elliptic Measures and Square Function Estimates on 1-Sided Chord-Arc Domains

Abstract

In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $$L^p$$ L p , for some finite p, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $$A_\infty $$ A ∞ . In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors–David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. In addition, we obtain that for two given elliptic operators $$L_1$$ L 1 and $$L_2$$ L 2 , the absolute continuity of the surface measure with respect to the elliptic measure of $$L_1$$ L 1 is equivalent to the same property for $$L_2$$ L 2 provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally, for the case on which $$L_2$$ L 2 is either the transpose of $$L_1$$ L 1 or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex.

On the condition for elliptic operators in 1-sided nontangentially accessible domains satisfying the capacity density condition

“…Proof of Theorem 6.1. We use some ideas from [9, Section 4] and [7,Section 4]. Let 𝑢 ∈ 𝑊 1,2 loc (Ω) ∩ 𝐿 ∞ (Ω) be a weak solution of 𝐿 1 𝑢 = 0 in Ω and assume that 𝑢 𝐿 ∞ (Ω) = 1.…”

“…Our next goal is to state a qualitative version of Theorem 1.1 in line with [7]. The 𝐴 ∞ condition will turn into absolute continuity.…”

On the $A_\infty$ condition for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

“…It is not hard to see that the sets {U ϑ Q,η 3 } Q∈D Q 0 have bounded overlap with constant depending on η. The following result was obtained in [8, Lemma 3.10] (for β > 0) and in [6,Lemma 3.40] (for β = 0), both in the context of 1-sided CAD, extending [40,Lemma 2.6] and [39,Lemma 2.3]. It is not hard to see that the proof works with no changes in our setting: Lemma 4.4.…”

“…Proof of Theorem 6.1. We use some ideas from [8, Section 4] and [6,Section 4]. Let u ∈ W 1,2 loc (Ω) ∩ L ∞ (Ω) be a weak solution of L 1 u = 0 in Ω and assume that u L ∞ (Ω) = 1.…”

“…Our next goal is to state a qualitative version of Theorem 1.2 in line with [6]. The A ∞ condition will turn into absolute continuity.…”