Quantitative stratification for some free-boundary problems

Edelen, Nick; Engelstein, Max

doi:10.1090/tran/7401

Cited by 24 publications

(48 citation statements)

References 15 publications

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“…The second claim of Theorem 1.1 (2) was proved in [21,Section 5.5] together with the Hausdorff dimension bound dim H (Sing 1 (∂Ω U )) ≤ d − d * , for d > d * , which follows by a dimension reduction argument based on the Weiss' monotonicity formula. The claim of Theorem 1.1 (2) was proved by Edelen and Engelstein in [11,Theorem 1.15] by a finer argument based on the quantitative dimension reduction of Naber and Valtorta [22,23]. We notice that [11] contains also a stratification result on Sing 1 (∂Ω U ).…”

Section: Introductionmentioning

confidence: 68%

“…The claim of Theorem 1.1 (2) was proved by Edelen and Engelstein in [11,Theorem 1.15] by a finer argument based on the quantitative dimension reduction of Naber and Valtorta [22,23]. We notice that [11] contains also a stratification result on Sing 1 (∂Ω U ).…”

Section: Introductionmentioning

confidence: 68%

“…A more refined argument in the spirit of Naber and Valtorta, essentially based on the Weiss' monotonicity formula and the structure of the blow-up limits, can be used to deduce that the set S j has finite (d − j)-dimensional Hausdorff measure. For more details on this technique in the context of the free-boundary problems considered in this paper we refer the reader to [11].…”

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confidence: 99%

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Regularity of the free boundary for the vectorial Bernoulli problem

Mazzoleni¹,

Terracini

Velichkov

2020

Analysis & PDE

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In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure D ⊂ R d , Λ > 0 andWe prove that, for any optimal vector U = (u 1 , . . . , u k ), the free boundary ∂(∪ k i=1 {u i = 0}) ∩ D is made of a regular part, which is relatively open and locally the graph of a C ∞ function, a (one-phase) singular part, of Hausdorff dimension at most d − d * , for a d * ∈ {5, 6, 7}, and by a set of branching (two-phase) points, which is relatively closed and of finite H d−1 measure. Our arguments are based on the NTA structure of the regular part of the free boundary.

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Section: Introductionmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 68%

mentioning

confidence: 99%

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Regularity of the free boundary for the vectorial Bernoulli problem

Mazzoleni¹,

Terracini

Velichkov

2020

Analysis & PDE

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“…22),(7.23), the monotonicity of I c.A/ .´i ; /,(7.21), the definition of U , and (7.16), we obtain the estimate (log 2/ ¡ ¡ t .B 2t .y//:…”

mentioning

confidence: 89%

“…Note that in [22], Edelen and Engelstein also studied the lower-dimensional strata of free boundary in various classes of free boundary problems by using the techniques of [34]. Their work is closer in spirit to [34,35], as they resort to the quantitative stratification of free boundary.…”

Section: Strategy Of the Proofmentioning

confidence: 99%

On the Singular Set of Free Interface in an Optimal Partition Problem

Alper

2019

Comm Pure Appl Math

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We study the singular set of free interface in an optimal partition problem for the Dirichlet eigenvalues. We prove that its upper (n − 2)‐dimensional Minkowski content, and consequently its (n − 2)‐dimensional Hausdorff measure, are locally finite. We also show that the singular set is countably (n − 2)‐rectifiable; namely, it can be covered by countably many C1‐manifolds of dimension (n − 2), up to a set of (n − 2)‐dimensional Hausdorff measure zero. Our results hold for optimal partitions on Riemannian manifolds and harmonic maps into homogeneous trees as well. © 2019 Wiley Periodicals, Inc.

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Minimizers for the Thin One‐Phase Free Boundary Problem

Engelstein

Kauranen

Prats

et al. 2021

Comm Pure Appl Math

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We consider the “thin one‐phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in ℝ+n+1 plus the area of the positivity set of that function in ℝn. We establish full regularity of the free boundary for dimensions n≤2, prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced by Alt and Caffarelli in 1981. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments that are less reliant on the underlying PDE. © 2021 Wiley Periodicals LLC.

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Quantitative stratification for some free-boundary problems

Cited by 24 publications

References 15 publications

Regularity of the free boundary for the vectorial Bernoulli problem

Regularity of the free boundary for the vectorial Bernoulli problem

On the Singular Set of Free Interface in an Optimal Partition Problem

Minimizers for the Thin One‐Phase Free Boundary Problem

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