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Regularity for free boundary surfaces minimizing degenerate area functionals

Carlo Gasparetto, Filippo Paiano, Bozhidar Velichkov

TL;DR

This work develops a sharp $\varepsilon$-regularity theory for free boundaries of almost-minimizers of a degenerate, weighted perimeter $\mathrm{Per}_{w_Ω}$ with weight $w_Ω(x)=\mathrm{dist}(x,\mathbb{R}^n\setminus\Omega)^a$, $a>0$. By combining an improvement-of-flatness scheme with interior and boundary Harnack inequalities and a weak viscosity framework, the authors show that near flat boundary points the boundary is a $C^{1,\gamma}$ graph meeting $\partial\Omega$ orthogonally, up to a singular set with controlled Hausdorff dimension. A key technical step is reducing the almost-minimizer problem to a viscosity solution of a linearized equation at blow-up and applying modern regularity results for degenerate weights. The results extend to Hölder weights under comparable growth conditions and illuminate Bernstein-type questions for weighted perimeters, providing a robust regularity theory for free boundaries in degenerate-weights settings.

Abstract

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to $d(\cdot,\mathbb{R}^n\setminusΩ)^a$ near $\partialΩ$ and $a>0$. This implies that the boundaries of almost-minimizers are $C^{1,γ_0}$-surfaces that touch $\partial Ω$ orthogonally, up to a Singular Set $\mathrm{Sing}(\partial E)$ whose Hausdorff dimension satisfies the bound $d_{\mathscr{H}}(\mathrm{Sing}(\partial E)) \leq n +a -(5+\sqrt{8})$.

Regularity for free boundary surfaces minimizing degenerate area functionals

TL;DR

This work develops a sharp -regularity theory for free boundaries of almost-minimizers of a degenerate, weighted perimeter with weight , . By combining an improvement-of-flatness scheme with interior and boundary Harnack inequalities and a weak viscosity framework, the authors show that near flat boundary points the boundary is a graph meeting orthogonally, up to a singular set with controlled Hausdorff dimension. A key technical step is reducing the almost-minimizer problem to a viscosity solution of a linearized equation at blow-up and applying modern regularity results for degenerate weights. The results extend to Hölder weights under comparable growth conditions and illuminate Bernstein-type questions for weighted perimeters, providing a robust regularity theory for free boundaries in degenerate-weights settings.

Abstract

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy , where is a weight asymptotic to near and . This implies that the boundaries of almost-minimizers are -surfaces that touch orthogonally, up to a Singular Set whose Hausdorff dimension satisfies the bound .

Paper Structure

This paper contains 20 sections, 34 theorems, 250 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem and assumptions on the boundary
  3. Main results
  4. Strategy of the proof
  5. Maximum Principles
  6. Regularity away from the boundary and optimality
  7. Outline of the paper
  8. Preliminaries and basic properties of almost-minimizers of Per w
  9. Regular domains and distance function
  10. Weighted perimeters
  11. Compactness
  12. Density estimates
  13. Variational and viscosity properties of (almost)-minimizers
  14. First variation and monotonicity formula for minimizers
  15. Viscosity properties
  16. ...and 5 more sections

Key Result

Theorem 1.3

There exist constants $\varepsilon_0, \lambda_0 > 0$ (small), $C_0>0$ (large) and $\gamma_0\in(0,1)$ depending only on $n, a, \alpha$, and $\beta$ such that the following holds. Let $\Omega$ be $\varkappa$-flat in the sense of ass:boundaryof_Omega, and let $E$ be a $(\vartheta,\beta)$-minimizer of $ for some $\nu\in\mathbb{S}^{{n-1}}$ with $\nu\perp e_n$ and $\varepsilon > 0$, and that Then, ther

Figures (7)

  • Figure 1: Two sets touching at $\partial\Omega$: before blow-up (left) and after (right)
  • Figure 2: Two sets touching at points in $\Omega$: before blow-up (left) and after (right)
  • Figure 3: The set $E_{j,R}$ constructed in the proof of \ref{['lemma:structure']}
  • Figure 4: A set $F$ touching $E$ from outside at $x_0\in\partial\Omega$. Notice that the assumption $E\subset\overline{\Omega}$ allows, in the smooth setting, that the tangent spaces at $x_0$ to $\partial E$ and $\partial F$ to differ when the touching point is at $\partial\Omega$
  • Figure 5: In red $F$, in blue $\{x_1 < \varphi - p\}$ and in yellow $E\setminus F$. The two dashed balls represent, respectively, $B_r(z)$ and $B_{cbr^2}(z)$. Three possible configurations based on the relation between $d_\Omega(z)$ and $r$ are represented.
  • ...and 2 more figures

Theorems & Definitions (44)

  • Definition 1.1: Almost-Minimizer of $\mathrm{Per}_{{w_{{\Omega}}}}$
  • Definition 1.2
  • Theorem 1.3: $\varepsilon$-Regularity
  • Proposition 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 1.8: Improvement of flatness
  • Lemma 2.1: Technical lemma on the distance function
  • Lemma 2.2
  • Definition 2.3: Sets of finite $w$-perimeter
  • ...and 34 more