A Variational Singular Perturbation Problem Motivated by Ericksen’s Model for Nematic Liquid Crystals

We study the asymptotic behavior, when $$\varepsilon \rightarrow 0$$ , of minimizers $$\{u_\varepsilon \}_{\varepsilon >0}$$ for energy $$\begin{aligned} E_\varepsilon (u)=\int _{\Omega }\Big (|\nabla u|^2+\big (\frac{1}{\varepsilon ^2}-1\big )|\nabla |u||^2\Big ), \end{aligned}$$ over class maps $$u\in H^1(\Omega ,{{\mathbb {R}}}^2)$$ satisfying boundary condition $$u=g$$ on $$\partial \Omega $$ where $$\Omega is a smooth, bounded and simply connected domain in $${{\mathbb {R}}}^2$$ $$g:\partial S^1$$ smooth data degree $$D\ge 1$$ . The motivation comes from simplified version Ericksen model nematic liquid crystals with variable orientation. prove convergence (up to subsequence) \}$$ towards singular $$S^1$$ –valued harmonic map $$u_*$$ result that resembles one obtained Bethuel et al. (Ginzburg–Landau Vortices, Birkhäuser, 1994) an analogous problem Ginzburg–Landau energy. There are however two striking differences between our involving First, problem, limit may have singularities of, strictly larger than one. Second, we find principle “equipartition” holds minimizers, i.e., contributions terms $$E_\varepsilon (u_\varepsilon )$$ essentially equal.