Wolff snowflakes

Lewis, John L.; Verchota, Gregory C.; Vogel, Andrew

doi:10.2140/pjm.2005.218.139

Cited by 19 publications

(28 citation statements)

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“…and ω Ω ⊥ H α for some α < d. The R 3 case is due to Wolff [Wol91] and the result for higher dimensions is due to Lewis, Verchota, and Vogel [LVV05]. A corollary of our main results, however, will show that, for NTA domains, mutual absolute continuity can only occur if α ≤ d.…”

Section: Introductionmentioning

confidence: 52%

Tangent measures and absolute continuity of harmonic measure

Azzam

Mourgoglou

2018

Rev. Mat. Iberoam.

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We show that for uniform domains Ω ⊆ R d+1 whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to α-dimensional Hausdorff measure unless α ≤ d. We employ a lemma that shows that, at almost every non-degenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure.2010 Mathematics Subject Classification. 31A15,28A75,28A78,28A33.

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Section: Introductionmentioning

confidence: 52%

Tangent measures and absolute continuity of harmonic measure

Azzam

Mourgoglou

2018

Rev. Mat. Iberoam.

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“…In this case we use a result from [8] to conclude that there exist Wolff snowflakes such that the sign of certain integral in the Wolff's program is independent of p when p is in an open interval containing 2. In order to show such a relation, in [8], Lewis, Nyström, and Vogel "perturb" off the p = 2 case from [12,9]. Note that results in [12,9] are valid only when n ≥ 3.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

On the absolute continuity of p-harmonic measure and surface measure in Reifenberg flat domains

Akman

2017

Pacific J. Math.

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In this paper, we study the set of absolute continuity of p-harmonic measure, µ, and (n − 1)−dimensional Hausdorff measure, H n−1 , on locally flat domains in R n , n ≥ 2. We prove that for fixed p with 2 < p < ∞ there exists a Reifenberg flat domain Ω ⊂ R n , n ≥ 2 with H n−1 (∂Ω) < ∞ and a Borel set K ⊂ ∂Ω such that µ(K) > 0 = H n−1 (K) where µ is the p-harmonic measure associated to a positive weak solution to p-Laplace equation in Ω with continuous boundary value zero on ∂Ω. We also show that there exists such a domain for which the same result holds when p is fixed with 2 − η < p < 2 for some η > 0 provided that n ≥ 3.This work is a generalization of a recent result of Azzam, Mourgoglou, and Tolsa when the measure µ is harmonic measure at x, ω = ω x , associated to the Laplace equation, i.e when p = 2.

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“…The following theorem proves that there are no Wolff snowflakes for which ω + and ω − are mutually absolutely continuous, answering a question in [16]. …”

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confidence: 90%

“…Here, whenever we refer to a "Wolff snowflake," we will mean a 2-sided NTA domain in R n , for which H − dim ω = n − 1. In [16], Lewis, Verchota and Vogel reexamined Wolff's construction and were able to produce "Wolff snowflakes" in R n , n ≥ 3, for which H − dim ω ± > n − 1, and others for which H − dim ω ± < n − 1. They also observed, as a consequence of the monotonicity formula in [1], that if ω…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, Lewis, Verchota and Vogel [16] conjectured that there are "Wolff snowflakes" in R n , n ≥ 3, with H − dim ω ± > n − 1, for which ω + , ω − are not mutually singular. In this paper we study these issues for domains which verify the weak regularity hypothesis of being 2-sided locally NTA (a condition which, of course, Wolff snowflakes verify).…”

Section: Introductionmentioning

confidence: 99%

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