[Paper Review] Quantitative stratification for some free-boundary problems

Nick Edelen, Max Engelstein|arXiv (Cornell University)|Feb 14, 2017

Advanced Mathematical Modeling in Engineering18 references4 citations

TL;DR

This paper establishes the rectifiability and uniform measure bounds for the singular set of the free boundary in the one-phase Alt-Caffarelli problem using quantitative stratification and the Rectifiable-Reifenberg framework. It proves that the $(n-k^*)$-dimensional Hausdorff measure of the singular set is uniformly bounded in compact subsets, with the bound depending only on $n$, $C^α$ norm of $Q$, its infimum, and geometric parameters of the domain.

ABSTRACT

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.

Motivation & Objective

• To establish the rectifiability and uniform packing bounds for the singular set of the free boundary in the one-phase Alt-Caffarelli problem.
• To extend the quantitative stratification framework to free boundary problems with non-constant coefficients.
• To demonstrate that the singular set has finite $(n-k^*)$-dimensional Hausdorff measure in compact subsets.
• To provide a robust framework applicable to broader classes of free boundary problems, including two-phase and vector-valued variants.
• To achieve effective dimension reduction through a corona-type decomposition and localized $L^2$-estimates on the density drop.

Proposed method

• Applies the quantitative stratification framework of Naber-Valtorta to classify points in the free boundary by their symmetry degree via blow-up analysis.
• Introduces $(k,\epsilon,r)$-strata as effective analogues of classical strata, capturing the degree of symmetry in blow-ups.
• Uses a corona-type decomposition with good and bad trees to cover the singular set with balls, ensuring uniform control on the sum of radii to the power $k^*$.
• Employs $L^2$-estimates on the density drop to control the oscillation of the frequency function and link it to the rectifiability of the singular set.
• Applies the Rectifiable-Reifenberg theorem to deduce rectifiability from Ahlfors-regularity and $L^2$-flatness estimates.
• Relies on the $\epsilon$-regularity theorem of Alt-Caffarelli and the monotonicity of Weiss' frequency function to control blow-up behavior.

Experimental results

Research questions

• RQ1 Can the singular set of the free boundary in the one-phase Alt-Caffarelli problem be shown to be rectifiable?
• RQ2 What is the sharp measure-theoretic size of the singular set in terms of Hausdorff dimension and measure?
• RQ3 Can the quantitative stratification and Rectifiable-Reifenberg framework be adapted to free boundary problems with non-constant coefficients?
• RQ4 How does the $k^*$-dimensional symmetry threshold relate to the rectifiability and measure bounds of the singular set?
• RQ5 Can effective packing estimates be derived for the singular set using localized geometric and analytic tools?

Key findings

• The $(n-k^*)$-dimensional Hausdorff measure of the singular set $\mathrm{sing}(u) \cap D$ is uniformly bounded by a constant $C$ depending only on $n$, $|Q|_{C^\alpha(D')}\!$, $\min_{D'} Q$, $\mathrm{dist}(D,\partial D')$, and $\mathcal{L}^n(D')$.
• The singular set $\mathrm{sing}(u) \cap D$ is $(n-k^*)$-rectifiable, meaning it is contained in a countable union of $C^1$ submanifolds up to $\mathcal{H}^{n-k^*}$-measure zero.
• The $\mathcal{H}^{n-k^*}$-measure of the $k^*$-stratum is upper-Ahlfors-regular away from the boundary, satisfying $\mu_{\nu}(B_r(x)) \leq c(n,\Lambda,\epsilon,\alpha) r^{n-k^*}$.
• The proof relies on a corona-type decomposition where the singular set is covered by balls from good and bad trees, with the total $r^{k^*}$-mass uniformly bounded.
• The framework applies to a broad class of related problems, including two-phase, vector-valued one-phase, and almost-minimizer problems with minimal modifications.
• The $L^2$-estimate on the density drop and the $\epsilon$-regularity theorem are instrumental in linking symmetry to rectifiability and measure bounds.