Compensation effects for anisotropic energies of two-dimensional unit vector fields

We study the highly anisotropic energy of two-dimensional unit vector fields given by \begin{align*} E_ε(u)= \int_Ω ($\mathrm{d}\mathrm{i}\mathrm{v}$,u)^2 + ε($\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}$,u)^2, dx,, \quad u\colonΩ\subset\mathbb R^2\to\mathbb S^1, \end{align*} in the limit $\epsilon \to 0$. This energy clearly loses control on the full gradient of $u$ as $\epsilon \to 0$, but, adapting tools from hyperbolic conservations laws, we show that it still controls derivatives of order 1/2. In particular, any bounded energy sequence $E_{\epsilon }(u_{\epsilon })\le C$ is compact in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{s,3}(\mathrm{\Omega })$ for $s<1\mathrm{/}2$. Moreover, this order 1/2 of differentiability is optimal, in the sense that any map $u\in W^{1\mathrm{/}2,4}(\mathrm{\Omega };{\mathbb{S}}^1)$ is a limit of a bounded energy sequence. We also establish compactness of boundary traces in $L^1(\mathrm{\partial }\mathrm{\Omega })$, and characterize the $\mathrm{\Gamma }$-limit in the simpler case of maps of a single variable and in the case of a thin-film model.