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

TL;DR

This paper investigates the anisotropic energy of 2D unit vector fields, showing it controls derivatives of order 1/2 in the limit, and characterizes the compactness and limits of energy sequences.

Contribution

It demonstrates that despite losing control of the full gradient, the energy still controls half-order derivatives and characterizes the limit behavior of sequences and boundary traces.

Findings

Energy controls derivatives of order 1/2 as epsilon approaches zero.

Bounded energy sequences are compact in fractional Sobolev spaces.

Characterization of the Gamma-limit for specific cases.

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.