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

Abstract

We study the highly anisotropic energy of two-dimensional unit vector fields given by \begin{align*} E_\epsilon(u)= \int_{\Omega} (\mathrm{div}\,u)^2 + \epsilon(\mathrm{curl}\,u)^2\, dx\,, \quad u\colon\Omega\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 $E_\epsilon(u_\epsilon)\leq C$ is compact in $W^{s,3}_{\mathrm{loc}}(\Omega)$ for $s<1/2$. Moreover, this order 1/2 of differentiability is optimal, in the sense that any map $u\in W^{1/2,4}(\Omega;\mathbb S^1)$ is a limit of a bounded energy sequence. We also establish compactness of boundary traces in $L^1(\partial\Omega)$, and characterize the $\Gamma$-limit in the simpler case of maps of a single variable and in the case of a thin-film model.