# 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} }

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.

#### 17 Citations

The Singular Strata of a Free-Boundary problem for harmonic measure

- Mathematics
- 2019

In this paper, we obtain quantitative estimates on the fine structure of the singular set of the mutual boundary ∂Ω for pairs of complementary domains, Ω,Ω ⊂ R which arise in a class of two-sided… Expand

Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries

- Mathematics
- 2021

We prove that for minimizers of the vectorial Alt-Caffarelli functional the two-phase singular set of the free boundary is rectifiable and the blow-up is unique almost everywhere on it. While the… Expand

On the existence of non-flat profiles for a Bernoulli free boundary problem

- Mathematics
- 2018

In this paper we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found… Expand

Minimizers for the Thin One‐Phase Free Boundary Problem

- Mathematics
- Communications on Pure and Applied Mathematics
- 2021

We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that… Expand

Estimates on the generalized critical strata of Green's function

- Mathematics
- 2019

In this paper, we obtain quantitative estimates on the fine structure of Green's functions for pairs of complementary domains, $\Omega^+, \Omega^- \subset \mathbb{R}^n$ which arise in a class of… Expand

Regularity of the free boundary for the two-phase Bernoulli problem

- Mathematics
- 2019

We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we… Expand

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

- Mathematics
- Communications on Pure and Applied Mathematics
- 2019

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… Expand

Unique Continuation on Convex Domains

- Mathematics
- 2019

In this paper, we adapt powerful tools from geometric analysis to get quantitative estimates on the quantitative strata of the generalized critical set of harmonic functions which vanish continuously… Expand

Stable cones in the thin one-phase problem.

- Physics, Mathematics
- 2020

The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem.
The problem of classifying stable (or minimal) homogeneous solutions in… Expand

Diameter and curvature control under mean curvature flow

- Mathematics
- 2017

Abstract:We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp… Expand

#### 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 $C^{1, \alpha}$

- Mathematics
- 1987

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. We… Expand

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 dimension… Expand

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 of… Expand

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 of… Expand

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 x… Expand

Discrete Reifenberg-type theorem

- Mathematics
- 2018

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 a… Expand

A Harnack inequality approach to the regularity of free boundaries part II: Flat free boundaries are Lipschitz

- Mathematics
- 1989

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) est… Expand

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