A variational analysis of nematic axisymmetric films: the covariant derivative case

TL;DR

This work rigorously analyzes the variational model for nematic axisymmetric films with covariant derivative on revolution surfaces spanning two coaxial rings. The authors reduce the two-dimensional energy to a one-dimensional functional Ec(rho) and prove the existence of minimizers for all c ≥ 0, along with detailed regularity and symmetry properties; in the zero-coupling limit (c=0) they recover a unique catenary minimizer under a geometric constraint, connecting to classical minimal surface theory. They establish a convexification that preserves or lowers energy, derive the Euler-Lagrange equations and a first integral for minimizers, and show that as c grows, minimizers become flatter and converge to the boundary radius r, effectively approaching a cylinder. The results lay a rigorous foundation for the covariant-derivative model of nematic surfaces and set the stage for extending to the full surface-gradient model and for numerical explorations of equilibrium shapes.

Abstract

Nematic surfaces are thin fluid structures, ideally two-dimensional, endowed with an in-plane nematic order. In 2012, two variational models have been introduced by Giomi [11] and by Napoli and Vergori [29,28]. Both penalize the area of the surface and the gradient of the director: in [11] the covariant derivative of the director is considered, while [28] deals with the surface gradient. In this paper, a complete variational analysis of the model proposed by Giomi is performed for revolution surfaces spanning two coaxial rings.

Figures (6)

• Figure 1.1: Three dimensional plots of $\varrho_c$ solution of \ref{['EL']} and $\varrho_0$ solution of \ref{['eq:catenaria']} having chosen $-h=-11/20, h=11/20$ and $r=9/10$. The red surface is the catenoid $\varrho_0$ while the grey surface is the minimizer $\varrho_c$ corresponding to the choice $c=0.1$.
• Figure 1.2: (a) Numerical solutions of \ref{['EL']} having chosen $-h= -1$, $h = 1$, as boundary conditions $\varrho(1) = \varrho(-1) = 7/2$ and $c = 0, 1,2,10,30$. The thicker blue line is the catenary for $c = 0$. (b) Numerical solutions of \ref{['EL']} having chosen $-h= -1$, $h = 1$, as boundary conditions $\varrho(1) = \varrho(-1) = 5$ and $c = 0, 1,2,10,30,100$. The thicker blue line is the catenary for $c = 0$.
• Figure 2.1: Graphical representation of the two piecewise affine functions $\varrho_1$ and $\varrho_2$ defined in \ref{['eq:rho1']} and \ref{['eq:rho2']}.
• Figure 3.1: In blue, the function $u(s)$ and in orange the unique abscissa $\Xi$ with $u(\Xi)=1$. Grey lines refer to the two abscissas $\xi_0$ and $\xi_1$ corresponding to the two catenaries. Green lines correspond to $u(\xi_0)$ and $u(\xi_1)$.
• Figure 3.2: In blue, the function $\mu(\xi)$ defined in \ref{['summaryfunction']} and in green a generic line at height $h/r$. Red lines refer to the two constants $1/m$ and $\omega$ introduced in Remark \ref{['rem:analisi_catenaria']} to distinguish the four possible cases. In the case $\frac{h}{r}<\frac{1}{m}$, the two intersection points $(\Pi_0/h, \mu(\Pi_0/h))$ and $(\Pi_1/h,\mu(\Pi_1/h))$, with $\Pi_0>\Pi_1$ are shown.

Theorems & Definitions (33)

• Theorem 1.1
• Proposition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 2.5
• proof
• proof : Proof of Proposition \ref{['lemmaconvexity-affine_con']}