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Uniqueness of capillary disks in three-dimensional domains

Henrique Nogueira Bastos

Abstract

We prove uniqueness results for capillary disks in three-dimensional domains that are modeled by an elliptic PDE, under the assumption that the domain admits a family of surfaces with suitable properties. Our main theorem generalizes Nitsche's result for capillary constant mean curvature disks in the Euclidean ball and is inspired by the extension of Hopf's uniqueness theorem for constant mean curvature spheres in Euclidean space due to Gálvez and Mira.

Uniqueness of capillary disks in three-dimensional domains

Abstract

We prove uniqueness results for capillary disks in three-dimensional domains that are modeled by an elliptic PDE, under the assumption that the domain admits a family of surfaces with suitable properties. Our main theorem generalizes Nitsche's result for capillary constant mean curvature disks in the Euclidean ball and is inspired by the extension of Hopf's uniqueness theorem for constant mean curvature spheres in Euclidean space due to Gálvez and Mira.

Paper Structure

This paper contains 10 sections, 7 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Strategy of the proof
  4. Boundary behavior
  5. Boundary conditions in adapted coordinates
  6. Asymptotic lines on the boundary
  7. Index of singularities and proof of the main result
  8. Principal lines near singularities
  9. Proof of the main theorem
  10. Applications

Key Result

Proposition 2.1

Let $\Sigma$ be an immersed capillary surface in $\Omega$ with contact angle $\alpha\in(0,\pi)$. Let $(u,v,z)$ be adapted coordinates to $(p,T_p\Sigma)$ for some $p\in \partial(\Sigma\cap \Omega)$, chosen so that $\{u=0\}$ corresponds to $\partial \Omega$ and $(0,v,z)$ are isothermal coordinates on for all $v$ sufficiently close to $0$.

Theorems & Definitions (18)

  • Definition 1.1: Transitive Capillary Family
  • Definition 1.2: Modeled Surface
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • ...and 8 more