A variational singular perturbation problem motivated by Ericksen's model for nematic liquid crystals

Abstract

We study the asymptotic behavior, when $\varepsilon\to0$, of the minimizers $\{u_\varepsilon\}_{\varepsilon>0}$ for the energy \begin{equation*} E_\varepsilon(u)=\int_{\Omega}\Big(|\nabla u|^2+\big(\frac{1}{\varepsilon^2}-1\big)|\nabla|u||^2\Big), \end{equation*} over the class of maps $u\in H^1(\Omega,{\mathbb R}^2)$ satisfying the boundary condition $u=g$ on $\partial\Omega$, where $\Omega$ is a smooth, bounded and simply connected domain in ${\mathbb R}^2$ and $g:\partial\Omega\to S^1$ is a smooth boundary data of degree $D\ge1$. The motivation comes from a simplified version of the Ericksen model for nematic liquid crystals with variable degree of orientation. We prove convergence (up to a subsequence) of $\{u_\varepsilon\}$ towards a singular $S^1$-valued harmonic map $u_*$, a result that resembles the one obtained in \cite{BBH} for an analogous problem for the Ginzburg-Landau energy. There are however two striking differences between our result and the one involving the Ginzburg-Landau energy. First, in our problem the singular limit $u_*$ may have singularities of degree strictly larger than one. Second, we find that the principle of \enquote{equi-partition} holds for the energy of the minimizers, i.e., the contributions of the two terms in $E_\varepsilon(u_\varepsilon)$ are essentially equal.