Existence and uniqueness of minimizers for axisymmetric nematic films

TL;DR

This work establishes the existence and uniqueness of minimizers for a one-dimensional reduction of the axisymmetric nematic film energy, where the director lies in the tangent plane of a revolution surface and the energy combines surface tension with the surface gradient nematic term. By a carefully designed relaxation in $W^{1,1}$ and a first-variation analysis, the authors prove existence of a unique minimizer $\varrho_c$ in a constrained class, and provide a complete geometric characterization: for $0<c<r^2$ the profile is convex and lies above the stable catenary, for $c=r^2$ it is constant, and for $c>r^2$ it is concave and approaches the asymptotic circle arc $\varrho_\infty$ as $c\to\infty$; in all cases they derive explicit $\varrho_c$ in terms of a parameter $E(c)$ solving a transcendental boundary condition. A key part of the analysis shows that $\varrho_c$ is monotone in $c$, and a Gamma-convergence perspective explains the limiting behavior as $c\to\infty$. Numerical simulations reinforce the theoretical picture, illustrating convex/concave transitions, convergence to the circumference, and the behavior of multiple critical points for small $c$. The results clarify the interplay between surface tension and nematic elasticity in axisymmetric thin films and provide a rigorous foundation for observed geometric patterns.

Abstract

Nematic surfaces are thin liquid films endowed with in-plane orientational order. We study a variational model in which the nematic director is constrained to lie in the tangent space of an axisymmetric surface, and the associated surface energy accounts for both surface tension and elastic nematic contributions. Here we adopt the surface gradient as the differential operator on the surface, we restrict our analysis to revolution surfaces spanning two coaxial rings, and we assume that the nematic director is aligned along parallels. In this setting, the energy functional reduces to a one-dimensional variational problem. We rigorously prove the existence and uniqueness of minimizers and we provide their complete geometric characterization. Finally, we run some numerical simulations.

Figures (3)

• Figure 5.1: Numerical representation of minimizers of $\mathcal{F}_c$: we choose $h=1$, as boundary conditions $\varrho_c(1) = \varrho_C(-1) = 7/2$ and from the bottom to the top $c = 0, 1,2,12.25(=r^2),30,100,1000$. The thicker blue line is the catenary for $c = 0$. The black line is the truncated circumference when $c\to\infty$, with equation $x^2+y^2=13.25=1/E_\infty^2$.
• Figure 5.2: A slice of revolution surfaces obtained from profiles that are minimizers of the energy functional $\mathcal{F}_c$. We choose $h=1$ and boundary conditions $\varrho_c(1) = \varrho_c(-1) = 7/2$. From the yellow surface to the blue one, we have the catenoid (yellow) for $c = 0$, the convex surface (orange) for $c = 5$, the cylinder (green) for $c= 12.25 = r^2$, the concave profile (red) for $c = 50$ and finally the region of a sphere generated by $\varrho_\infty$ in blue.
• Figure 5.3: Qualitative study of the number of solutions of \ref{['trascr']} for different values of $c$. From the bottom to the top, (fixing $r=1$) we have $h=H(E)=0.1, 0.4, \omega\sim 0.528, 1/m\sim 0.6627$ and $c=0,0.0005,0.002, 0.005, c^*\sim 0.0257, 0.05$. $\overline{E}$ is the abscissa of the maximum of $H(E)$ for $c=0$. An horizontal line determines two intersections with $H(E)$ in the case $c=0$, three intersections when $c\in(0,c^*)$. If $c\geq c^*$, there is a unique intersection corresponding to the minimizer of the energy functional $\mathcal{F}_c$.

Theorems & Definitions (31)

• Theorem 1.1
• Proposition 1
• Remark 1
• Theorem 2.1: bm1991
• Theorem 2.2: bm1991
• Lemma 1
• proof
• Remark 2
• Lemma 2
• proof