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Varifolds with capillary boundary

Guofang Wang, Xuwen Zhang

TL;DR

This work develops a unified measure-theoretic framework for capillary phenomena in arbitrary dimensions and codimensions by introducing varifolds with capillary boundary via a Neumann-type condition on the capillary bundle $G_{m,\beta}(S)$. It establishes key analytic tools—monotonicity formulas, boundary rectifiability, tangent-cone classification, and integral compactness—and connects these to curvature varifolds and Brakke flow with capillary boundary. The results extend classical Allard boundary theory to capillary settings beyond codimension one and provide a robust compactness theory for capillary variational problems, enabling weak formulations and stability analyses. The framework equips curvature-varifold and Brakke-flow approaches with capillary boundary data, offering new avenues for existence, regularity, and evolution of capillary submanifolds in higher codimensions.

Abstract

In this paper we introduce and study a new class of varifolds in $\mathbf{R}^{n+1}$ of arbitrary dimensions and co-dimensions, which satisfy a Neumann-type boundary condition characterizing capillarity. The key idea is to introduce a Radon measure on a subspace of the trivial Grassmannian bundle over the supporting hypersurface as a generalized boundary with prescribed angle, which plays a role as a measure-theoretic capillary boundary. We show several structural properties, monotonicity inequality, boundary rectifiability, classification of tangent cones, and integral compactness for such varifolds under reasonable conditions. This Neumann-type boundary condition fits very well in the context of curvature varifold and Brakke flow, which we also discuss.

Varifolds with capillary boundary

TL;DR

This work develops a unified measure-theoretic framework for capillary phenomena in arbitrary dimensions and codimensions by introducing varifolds with capillary boundary via a Neumann-type condition on the capillary bundle . It establishes key analytic tools—monotonicity formulas, boundary rectifiability, tangent-cone classification, and integral compactness—and connects these to curvature varifolds and Brakke flow with capillary boundary. The results extend classical Allard boundary theory to capillary settings beyond codimension one and provide a robust compactness theory for capillary variational problems, enabling weak formulations and stability analyses. The framework equips curvature-varifold and Brakke-flow approaches with capillary boundary data, offering new avenues for existence, regularity, and evolution of capillary submanifolds in higher codimensions.

Abstract

In this paper we introduce and study a new class of varifolds in of arbitrary dimensions and co-dimensions, which satisfy a Neumann-type boundary condition characterizing capillarity. The key idea is to introduce a Radon measure on a subspace of the trivial Grassmannian bundle over the supporting hypersurface as a generalized boundary with prescribed angle, which plays a role as a measure-theoretic capillary boundary. We show several structural properties, monotonicity inequality, boundary rectifiability, classification of tangent cones, and integral compactness for such varifolds under reasonable conditions. This Neumann-type boundary condition fits very well in the context of curvature varifold and Brakke flow, which we also discuss.

Paper Structure

This paper contains 36 sections, 40 theorems, 274 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Boundary rectifiability
  4. Classification of tangent cones
  5. Integral compactness
  6. Comparison with the geometric variational formulation
  7. Curvature varifolds with capillary boundary
  8. Brakke flow with capillary boundary
  9. Allard-type regularity
  10. Brief outline of the paper
  11. Preliminaries
  12. Varifolds
  13. Rectifiable varifolds
  14. Allard's varifolds with boundary
  15. Prescribed contact angle condition
  16. ...and 21 more sections

Key Result

Theorem 1.4

Let $\Omega\subset\mathbf{R}^{n+1}$ be a bounded domain of class $C^2$ and $\beta\in C^1(S,(0,\pi))$. Let $V\in{\bf V}^m_\beta(\Omega)$ with boundary varifold $\Gamma$, such that $\mathbf{H}\in L^p(\mu_V)$ for some $p\in(m,\infty)$, and $\Theta^m(\mu_V,x)\geq a>0$ for $\mu_V$-a.e. $x$. Then for any

Figures (2)

  • Figure 1: Unions of two planes in $G_{n,\beta_0}(\partial\mathbf{R}^{n+1}_+)$ with common boundary
  • Figure 2: Unions of two planes in $G_{n,\beta_0}(\partial\mathbf{R}^{n+1}_+)$ with distinct boundaries

Theorems & Definitions (99)

  • Definition 1.1: Capillary bundle
  • Definition 1.2: Varifolds with capillary boundary
  • Definition 1.3: Capillary boundary point
  • Theorem 1.4: Boundary rectifiability
  • Definition 1.5: Varifolds with rectifiable capillary boundary
  • Theorem 1.6: Classification of tangent cones
  • Theorem 1.7: Integral compactness
  • Definition 1.8: Geometric variational formulation
  • Definition 1.9: Curvature varifolds with capillary boundary
  • Theorem 1.10: Compactness for curvature varifolds
  • ...and 89 more