# Quantitative stratification for some free-boundary problems

@article{Edelen2018QuantitativeSF,
title={Quantitative stratification for some free-boundary problems},
author={Nick Edelen and Max Engelstein},
journal={Transactions of the American Mathematical Society},
year={2018}
}
• Published 14 February 2017
• Mathematics
• Transactions of the American Mathematical Society
In this paper we prove the rectifiability of and measure bounds on the singular set of the free boundary for minimizers of a functional first considered by Alt-Caffarelli. Our main tools are the Quantitative Stratification and Rectifiable-Reifenberg framework of Naber-Valtorta, which allow us to do a type of "effective dimension-reduction." The arguments are sufficiently robust that they apply to a broad class of related free boundary problems as well.
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#### References

SHOWING 1-10 OF 29 REFERENCES
Some remarks on stability of cones for the one-phase free boundary problem
• Mathematics
• 2014
We show that stable cones for the one-phase free boundary problem are hyperplanes in dimension 4. As a corollary, both one and two-phase energy minimizing hypersurfaces are smooth in dimension 4.
A Harnack Inequality Approach to the Regularity of Free Boundaries. Part I: Lipschitz Free Boundaries are $C^{1, \alpha}$
This is the first in a series of papers where we intend to show, in several steps, the existence of classical (or as classical as possible) solutions to a general two-phase free-boundary system. WeExpand
A singular energy minimizing free boundary
• Mathematics
• 2009
Abstract We consider the problem of minimizing the energy functional ∫(|∇u|2 + χ {u>0}). We show that the singular axissymmetric critical point of the functional is an energy minimizer in dimensionExpand
Rectifiability and upper Minkowski bounds for singularities of harmonic $Q$-valued maps
• Mathematics
• Commentarii Mathematici Helvetici
• 2018
In this article we prove that the singular set of Dirichlet-minimizing $Q$-valued functions is countably $(m-2)$-rectifiable and we give upper bounds for the $(m-2)$-dimensional Minkowski content ofExpand
Stratification for the singular set of approximate harmonic maps
• Mathematics
• 2016
AbstractThe aim of this note is to extend the results in Naber and Valtorta (Ann Math (2) 185:131–227, https://doi.org/10.4007/annals.2017.185.1.3, 2017) to the case of approximate harmonic maps.Expand
Reifenberg Parameterizations for Sets with Holes
• Mathematics
• 2009
We extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization ofExpand
Free boundary regularity for almost-minimizers
• Physics, Mathematics
• Advances in Mathematics
• 2019
Abstract In this paper we study the free boundary regularity for almost-minimizers of the functional J ( u ) = ∫ Ω | ∇ u ( x ) | 2 + q + 2 ( x ) χ { u > 0 } ( x ) + q − 2 ( x ) χ { u 0 } ( x ) d xExpand
Discrete Reifenberg-type theorem
The paper proves that a bound on the averaged Jones' square function of a measure implies an upper bound on the measure. Various types of assumptions on the measure are considered. The theorem is aExpand
A Harnack inequality approach to the regularity of free boundaries part II: Flat free boundaries are Lipschitz
Soit u une solution faible d'un probleme aux limites libres du type Δu=0 sur Ω + ={u>0} et sur Ω − ={u≤0}° et soit le long de la frontiere libre, F=∂Ω + , la relation u v + =G(uν − , X, ν) estExpand
Gradient estimates for variable coefficient parabolic equations and singular perturbation problems
• Mathematics
• 1998
In this article we prove, via monotonicity formulas, interior and boundary gradient estimates for solutions to second order parabolic equations, in divergence form, with Dini top order coefficients.Expand