Abstract:
Several authors (Alama, Bronsard, Golovaty, Mironescu) have recently studied
the minimizers $\{u_\varepsilon\}$ of the energy $ E_\varepsilon(u) = E^{g,\alpha}_\varepsilon(u)$, given by
\begin{equation}
\label{eq:1}
E_\varepsilon(u) = \frac{1}{2} \int_\Omega \left( |\nabla
u|^2 + \frac{1}{2\varepsilon^2}(1 - |u|^2)^2 \right) \, dx + \frac{1}{2\varepsilon^s} \int_{\partial\Omega}
{\widetilde W}(u,g) \, ds
\end{equation}
over $H^1(\Omega, \mathbb{C})$, where
\begin{equation}
\label{eq:2}
{\widetilde W}(u,g) = \frac{1}{2}(|u|^2 - 1)^2 + [(u, g) - \cos\alpha]^2
\end{equation}
with $\alpha \in (0, \pi/2)$ and $s \in (0,1)$, and where
$g : \partial\Omega \to S^1$ is a smooth function of
degree $D \ge 1$. The motivation comes from a thin-film limit of the
Landau-de Gennes energy for a nematic liquid crystal. For
$s < \frac{1}{2\big((\alpha/\pi)^2+(1-\alpha/\pi)^2\big)}$, it was shown that in the limit $\varepsilon \to 0$, boundary
defects (called “boojums”) appear.
\par However, the first term in
\eqref{eq:2} is not part of the physical model, but rather added
artificially to the energy for “technical reasons”.
There are two main goals of our work:
- To show that the results obtained for the energy
\eqref{eq:1} remain valid for the energy derived from the physical
model, namely with $W(u,g) = [(u, g) - \cos\alpha]^2$, rather than $ {\widetilde W}(u,g)$
as defined in \eqref{eq:2}.
- To obtain more precise information on the minimizers
$\{u_\varepsilon\}$ as $\varepsilon \to 0$ in the "boojums regime"; e.g., to show that
$|u_\varepsilon| \to 1$ uniformly on $\bar\Omega$.
This is a joint work with Dmitry Golovaty.
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