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Allard-Type Regularity for Varifolds with Prescribed Contact Angle

Gaoming Wang

Abstract

Given a bounded $C^2$ domain in $\mathbb{R}^{n+1}$ and an integral $n$-rectifiable varifold $V$ with bounded first variation and bounded generalized mean curvature. Given a $C^1$ function $θ$ defined on the boundary of the domain with range $(0,π)$, we assume $V$ has prescribed contact angle $θ$ with $\partial Ω$ and the tangent cone of $V$ at a point $X \in \partial Ω$ is a half-hyperplane of density one. Then we can show that the support of $V$ is a $C^{1,γ}$ hypersurface with boundary near $X$ for some $γ\in (0,1)$.

Allard-Type Regularity for Varifolds with Prescribed Contact Angle

Abstract

Given a bounded domain in and an integral -rectifiable varifold with bounded first variation and bounded generalized mean curvature. Given a function defined on the boundary of the domain with range , we assume has prescribed contact angle with and the tangent cone of at a point is a half-hyperplane of density one. Then we can show that the support of is a hypersurface with boundary near for some .
Paper Structure (17 sections, 49 theorems, 319 equations, 2 figures)

This paper contains 17 sections, 49 theorems, 319 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Theorem
  3. Basic Notation
  4. Measures and Varifolds
  5. Main Results
  6. $\mu$-stationary quadruple
  7. $\mu$-flat metric
  8. Varifolds with prescribed contact angle
  9. Estimation of $L^2$ excess
  10. Properties of Blow-ups
  11. Improvement of excess
  12. Free boundary case
  13. Proof of Main Results
  14. Almost Stationary Varifolds under $C^1$ Metrics
  15. Theory of Varifolds under $C^1$ Metrics
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose $V$ is the integral $n$-rectifiable varifold described above with prescribed contact angle $\theta$, and suppose $\boldsymbol{H}$ is uniformly bounded. If at a point $X \in \mathrm{spt}\|V\| \cap \partial \Omega$, $V$ has a multiplicity-one tangent half-hyperplane at $X$, then the support of

Figures (2)

  • Figure 1: The stationary varifold after reflection
  • Figure 2: Choice of $U$ and $\gamma$

Theorems & Definitions (119)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Definition 3.1
  • Proposition 3.2
  • ...and 109 more