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Geometric criteria for the existence of capillary surfaces in tubes

Giorgio Saracco

Abstract

We review some geometric criteria and prove a refined version, that yield existence of capillary surfaces in tubes $Ω\times \mathbb{R}$ in a gravity free environment, in the case of physical interest, that is, for bounded, open, and simply connected $Ω\subset \mathbb{R}^2$. These criteria rely on suitable weak one-sided bounds on the curvature of the boundary of the cross-section $Ω$.

Geometric criteria for the existence of capillary surfaces in tubes

Abstract

We review some geometric criteria and prove a refined version, that yield existence of capillary surfaces in tubes in a gravity free environment, in the case of physical interest, that is, for bounded, open, and simply connected . These criteria rely on suitable weak one-sided bounds on the curvature of the boundary of the cross-section .
Paper Structure (5 sections, 2 theorems, 31 equations, 3 figures)

This paper contains 5 sections, 2 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A related problem
  3. Convex sets
  4. Non convex piecewise Lipschitz sets
  5. A refined criterion

Key Result

Theorem 2.1

Let $\Omega$ be a bounded, open, and simply connected subset of $\mathbb{R}^2$ satisfying eq:P=H--eq:Poincare, and let $H\in \mathbb{R}$ fixed. The following are equivalent:

Figures (3)

  • Figure 1: Two solutions of the prescribed curvature equation in $1$d. On the left one without enforcing any boundary condition, while on the right with a Neumann-like condition.
  • Figure 2: A set $\Omega$ that, when smoothed out, satisfies the curvature condition \ref{['eq:curvature_condition']}, yet existence of solutions of the PDE \ref{['eq:cap']} with boundary condition \ref{['eq:bc']} for $\gamma=0$ fails.
  • Figure 3: The Pinocchio set.

Theorems & Definitions (7)

  • Theorem 2.1
  • proof
  • Definition 4.1
  • Definition 4.2
  • Definition 5.1
  • proof
  • Corollary 5.3