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Improved regularity for minimizing capillary hypersurfaces

Otis Chodosh, Nick Edelen, Chao Li

TL;DR

This work establishes improved regularity for minimizing capillary hypersurfaces by proving that their singular set has codimension at least $4$, with sharper bounds when the capillary angle is near $0$, $\frac{\pi}{2}$, or $\pi$. A central idea is to connect the capillary problem to the one-phase Alt-Caffarelli free boundary problem via a rigorous small-angle expansion, showing that blow-ups converge to Alt-Caffarelli minimizers; in low dimensions these minimizers are linear, yielding flat capillary cones. The authors treat the near-zero-angle regime through rigorous AC-type convergence and Hodograph-transform techniques, including curvature bounds and a priori estimates, while the near-$\frac{\pi}{2}$ regime in dimensions $4\le n\le 6$ is handled by Simon-type identities and trace inequalities to deduce flatness. Collectively, the results deepen boundary regularity theory for capillary hypersurfaces and illuminate the deep link between capillary geometry and classic free boundary methods, with potential applications in comparison geometry and minimal surface theory.

Abstract

We give improved estimates for the size of the singular set of minimizing capillary hypersurfaces: the singular set is always of codimension at least $4$, and this estimate improves if the capillary angle is close to $0$, $\fracπ{2}$, or $π$. For capillary angles that are close to $0$ or $π$, our analysis is based on a rigorous connection between the capillary problem and the one-phase Bernoulli problem.

Improved regularity for minimizing capillary hypersurfaces

TL;DR

This work establishes improved regularity for minimizing capillary hypersurfaces by proving that their singular set has codimension at least , with sharper bounds when the capillary angle is near , , or . A central idea is to connect the capillary problem to the one-phase Alt-Caffarelli free boundary problem via a rigorous small-angle expansion, showing that blow-ups converge to Alt-Caffarelli minimizers; in low dimensions these minimizers are linear, yielding flat capillary cones. The authors treat the near-zero-angle regime through rigorous AC-type convergence and Hodograph-transform techniques, including curvature bounds and a priori estimates, while the near- regime in dimensions is handled by Simon-type identities and trace inequalities to deduce flatness. Collectively, the results deepen boundary regularity theory for capillary hypersurfaces and illuminate the deep link between capillary geometry and classic free boundary methods, with potential applications in comparison geometry and minimal surface theory.

Abstract

We give improved estimates for the size of the singular set of minimizing capillary hypersurfaces: the singular set is always of codimension at least , and this estimate improves if the capillary angle is close to , , or . For capillary angles that are close to or , our analysis is based on a rigorous connection between the capillary problem and the one-phase Bernoulli problem.
Paper Structure (15 sections, 21 theorems, 125 equations, 1 figure)

This paper contains 15 sections, 21 theorems, 125 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Relationship with one-phase free boundary problem
  3. Organization
  4. Acknowledgements
  5. Preliminaries
  6. Notation
  7. First and second variation of capillary functional
  8. The Alt-Caffarelli functional
  9. The case $n=3$
  10. The case $n=4$ with $\theta$ close to $0$
  11. Preliminaries
  12. Inhomogenous blow-up to Alt-Caffarelli
  13. A priori estimates
  14. Proof of Theorems \ref{['thm:graphical.convergence.to.AC.minimizer']}, \ref{['thm:AC.bernstein.implies.capillary.bernstein']}, \ref{['thm:general-blow-up']}
  15. The case $4\leqslant n \leqslant 6$ with $\theta$ close to $\pi/2$

Key Result

Theorem 1.1

There exist $\varepsilon_0, \varepsilon_1>0$ such that the following holds. Let $(X^{n+1},h)$ be a smooth manifold with boundary, $\sigma\in C^1(\partial X)$, $-1<\sigma<1$, and $g\in C^1(X)$. Suppose that $E\subset X$ is a Caccioppoli set that minimizes eq:capillary.functional. Then $M=\mathop{\mat

Figures (1)

  • Figure :

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2: JerisonSavin
  • proof
  • Proposition 3.1
  • proof
  • ...and 33 more