Local Set Approximation: Mattila–vuorinen Type Sets, Reifenberg type Sets, and Tangent Sets

Badger, Matthew; Lewis, Stephen A.

doi:10.1017/fms.2015.26

Cited by 12 publications

(28 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As our conditions are reminiscent of those introduced by Reifenberg [Rei60], we often refer to these sets as Reifenberg type sets which are well approximated by zero sets of harmonic polynomials. This class of sets plays a crucial role in the study of a two-phase free boundary problem for harmonic measure with weak initial regularity, examined first by Kenig and Toro [KT06] and subsequently by Kenig, Preiss and Toro [KPT09], Badger [Bad11,Bad13], Badger and Lewis [BL15], and Engelstein [Eng16]. Our results are partly motivated by several open questions about the structure and size of the singular set in the free boundary, which we answer definitively below.…”

Section: Introductionmentioning

confidence: 66%

“…The improved dimension bounds on A \ A 1 in Theorem 1.8 require a refinement of (1.4) for Σ p ∈ H n,d that separate R n into complementary NTA domains, whose existence was postulated in [BL15, Remark 9.5]. Using the quantitative stratification machinery introduced in [CNV15], we demonstrate that near its singular points a zero set Σ p ∈ H n,d with the separation property does not resemble Σ h × R n−2 for any Σ h ∈ F 2,k , 2 ≤ k ≤ d. This leads us to a version of (1.4) with right hand side C(n, d, ε)r 3−ε for all ε > 0 and thence to dim M A \ A 1 ≤ n − 3 using [BL15]. In addition, we show that at "even degree" singular points, a zero set Σ p with the separation property, does not resemble Σ h × R n−3 for any Σ h ∈ F 3,2k , 2 ≤ 2k ≤ d. This leads us to the bound dim H Γ 2 ∪ Γ 4 ∪ · · · ≤ n − 4.…”

Section: Introductionmentioning

confidence: 99%

“…In the present setting, the role of the model sets S is played by H n,d . For background on the theory of local set approximation and summary of results from [BL15], we refer the reader to Appendix A. The core geometric result at the heart of Theorem 1.1 is the following property of zero sets of harmonic polynomials: H n,k points can be detected in zero sets of harmonic polynomials of degree d (1 ≤ k ≤ d) by finding a single, sufficiently good approximation at a coarse scale.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Structure of sets which are well approximated by zero sets of harmonic polynomials

2017

Self Cite

View full text Add to dashboard Cite

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how "degree k points" sit inside zero sets of harmonic polynomials in R n of degree d (for all n ≥ 2 and 1 ≤ k ≤ d) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of sets, including sharp Hausdorff and Minkowski dimension estimates on the singular set of "degree k points" (k ≥ 2) without proving uniqueness of blowups or aid of PDE methods such as monotonicity formulas. In addition, we show that in the presence of a certain topological separation condition, the sharp dimension estimates improve and depend on the parity of k. An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro. 18 6. Dimension bounds in the presence of good topology 20 7. Boundary structure in terms of interior and exterior harmonic measures 27 Appendix A. Local set approximation 29 Appendix B. Limits of complimentary NTA domains 35 References

show abstract

Section: Introductionmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Structure of sets which are well approximated by zero sets of harmonic polynomials

2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…Thus, sets that are locally bilaterally well approximated by S are like Reifenberg sets with vanishing constants (e.g. see [KT99]), except with approximation by hyperplanes replaced by approximation by sets in S. See [BL15] for further discussion. Using this terminology, Theorem 2.6 says that the pseudotangent sets of the boundary of a two-sided NTA domain are boundaries of unbounded two-sided NTA domains.…”

Section: Blowups Of Harmonic Measure On Nta Domainsmentioning

confidence: 99%

“…(i) There exist d 0 ≥ 1 depending on at most n and the NTA constants of Ω + and Ω − such that ∂Ω is locally bilaterally well approximated (in a Reifenberg sense) by zero sets of harmonic polynomials p : [BET17] (also see [BL15]). In fact, we proved in [BET17] that (ii) and (iii) hold on any closed set satisfying the bilateral approximation in (i).…”

Section: Introductionmentioning

confidence: 99%

Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data

2020

Self Cite

View full text Add to dashboard Cite

In non-variational two-phase free boundary problems for harmonic measure, we examine how the relationship between the interior and exterior harmonic measures of a domain Ω ⊂ R n influences the geometry of its boundary. This type of free boundary problem was initially studied by Kenig and Toro in 2006 and was further examined in a series of separate and joint investigations by several authors. The focus of the present paper is on the singular set in the free boundary, where the boundary looks infinitesimally like zero sets of homogeneous harmonic polynomials of degree at least 2. We prove that if the Radon-Nikodym derivative of the exterior harmonic measure with respect to the interior harmonic measure has a Hölder continuous logarithm, then the free boundary admits unique geometric blowups at every singular point and the singular set can be covered by countably many C 1,β submanifolds of dimension at most n − 3. This result is partly obtained by adapting tools such as Garofalo and Petrosyan's Weiss type monotonicity formula and an epiperimetric inequality for harmonic functions from the variational to the non-variational setting.

show abstract

A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

Nimer

2017

Calc. Var.

View full text Add to dashboard Cite

The study of the geometry of n-uniform measures in R d has been an important question in many fields of analysis since Preiss' seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski-Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension n − 3.In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most n − 3.A measure Φ is n-rectifiable if it is absolutely continuous to H n and there exists a countable collection of C 1 n-manifolds {M j } j such that Φ(R d \ j M j ) = 0. In [P], Preiss proved the following remarkable theorem relating the rectifiability of a measure to its density.Theorem 1.1 ([P]). A Radon measure Φ of R d is n-rectifiable if and only if it satisfies the following property:The density Θ n (x) = lim r→0 Φ(B(x, r)) ω n r n exists and is positive andTo prove this theorem, Preiss studies the geometry of n-uniform measures which appear as tangents (blow-ups) to measures satisfying (1.2). A measure µ is said to be n-uniform if there exists a constant c > 0 such that for any x in the support of µ and any radius r > 0, we have:( 1.3) In [P], Preiss also provides a classification of the cases n = 1, 2 in R d for any d. In these cases, µ is n-Hausdorff measure restricted to a line or a plane respectively. Interestingly, flat measures are not the only examples of uniform measures. Indeed, in [KoP], Kowalski and Preiss proved that µ is (d − 1)-uniform in R d if and only if µ = H d−1 V where V is a (d − 1)-plane, or d ≥ 4 and there exists an orthonormal system of coordinates in which µ = H d−1 (C × W ) where W is a (d − 4)-plane and C is the KP-cone. The classification for n ≥ 3 and codimension d − n ≥ 2 remains an open question.Kirchheim and Preiss later proved in [KiP] that the support Σ of a uniformly distributed measure (of which n-uniform measures are an example) is a real analytic variety, namely the intersection of countably many zero sets of analytic functions. An application of the stratification theorem for real analytic varieties implies that Σ must be a countable union of real analytic manifolds and the singular set has Hausdorff dimension at most (n − 1).We investigate the Hausdorff dimension of the singular set S µ of an n-uniform measure µ. Our main result is the following theorem.Theorem 1.2. Let µ be an n-uniform measure in R d , 3 ≤ n ≤ d. Then dim H (S µ ) ≤ n − 3.In the cases n = 3, d = 3, it is a standard result that the only 3-uniform measure (up to normalization) is 3-Lebesgue measure. In this case, the bound is obvious. To see that this bound cannot be improved, let n ≥ 3, d > n and consider the measure µ defined as:where M is the set M = (x 1 , . . . , x d ); x 2 4 = x 2 1 + x 2 2 + x 2 3 and x n+1 = . ...

show abstract

Local Set Approximation: Mattila–vuorinen Type Sets, Reifenberg type Sets, and Tangent Sets

Cited by 12 publications

References 34 publications

Structure of sets which are well approximated by zero sets of harmonic polynomials

Structure of sets which are well approximated by zero sets of harmonic polynomials

Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data

A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

Contact Info

Product

Resources

About