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Compactness of capillary hypersurfaces with mean curvature prescribed by ambient functions

Xuwen Zhang

TL;DR

This work studies compactness for capillary hypersurfaces whose mean curvature is prescribed by ambient functions. It develops oriented integral curvature varifolds with capillary boundary by extending Wang–Zhang’s unoriented capillary framework to the oriented setting and proves a Hutchinson-type compactness result under natural area, curvature, and boundary bounds. The limit object retains a prescribed contact angle along a boundary measure and decomposes into interior and capillary boundary components, with a clear description of potential hidden boundaries driven by the nodal set of the ambient mean curvature. A second main result extends the framework to capillary hypersurfaces whose boundary mean curvature is prescribed along the boundary manifold, under stronger curvature bounds, yielding a consistent capillary prescribed boundary mean curvature theory. Collectively, the results provide a rigorous variational framework for capillary mean curvature problems with ambient data and boundary constraints, enhancing understanding of boundary cancellations and regularity in the capillary setting.

Abstract

We prove a compactness result for capillary hypersurfaces with mean curvature prescribed by ambient functions, which generalizes the results of Schätzle and Bellettini to the capillary case. The proof relies on extending the definition of (unoriented) curvature varifolds with capillary boundary introduced by Wang-Zhang to the context of oriented integral varifolds. We also discuss the case when the mean curvature of the boundary is prescribed.

Compactness of capillary hypersurfaces with mean curvature prescribed by ambient functions

TL;DR

This work studies compactness for capillary hypersurfaces whose mean curvature is prescribed by ambient functions. It develops oriented integral curvature varifolds with capillary boundary by extending Wang–Zhang’s unoriented capillary framework to the oriented setting and proves a Hutchinson-type compactness result under natural area, curvature, and boundary bounds. The limit object retains a prescribed contact angle along a boundary measure and decomposes into interior and capillary boundary components, with a clear description of potential hidden boundaries driven by the nodal set of the ambient mean curvature. A second main result extends the framework to capillary hypersurfaces whose boundary mean curvature is prescribed along the boundary manifold, under stronger curvature bounds, yielding a consistent capillary prescribed boundary mean curvature theory. Collectively, the results provide a rigorous variational framework for capillary mean curvature problems with ambient data and boundary constraints, enhancing understanding of boundary cancellations and regularity in the capillary setting.

Abstract

We prove a compactness result for capillary hypersurfaces with mean curvature prescribed by ambient functions, which generalizes the results of Schätzle and Bellettini to the capillary case. The proof relies on extending the definition of (unoriented) curvature varifolds with capillary boundary introduced by Wang-Zhang to the context of oriented integral varifolds. We also discuss the case when the mean curvature of the boundary is prescribed.

Paper Structure

This paper contains 18 sections, 11 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Capillary boundary condition
  3. Main results
  4. Outline of the paper
  5. Acknowledgements
  6. Preliminaries
  7. Oriented integral varifolds
  8. Induced currents
  9. Induced unoriented varifolds
  10. Hutchinson's compactness theorem
  11. Prescribed contact angle condition
  12. Oriented integral curvature varifolds with capillary boundary
  13. Definition and basic properties
  14. Unoriented integral curvature varifolds with capillary boundary
  15. Proof of the main results
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $\Omega\subset\mathbf{R}^{n+1}$ be a bounded domain of class $C^2$, $S\coloneqq\partial\Omega$ with $\nu^S$ the inner unit normal to $\Omega$ along $S$, $\beta\in C^1(S, (0,\pi))$. For $l\in\mathbb{N}$, let $E_l\subset\Omega$ be open subset with $D_l\coloneqq\partial E_l\cap\partial\Omega$ and $ let $g_l,g:\overline\Omega\rightarrow\mathbf{R}$ be functions in $C^0(\overline\Omega)$ such that t

Theorems & Definitions (29)

  • Definition 1.1: Oriented capillary bundle
  • Definition 1.2: Oriented integral curvature varifolds with capillary boundary
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 2.1: Disintegration
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4: Unoriented capillary bundle
  • ...and 19 more