We study the highly anisotropic energy of two-dimensional unit vector fields given by align* E_ε (u) = _Ω (div\, u) ² + ε (curl\, u) ²\, dx\, , uΩ R² S¹\, align* in the limit ε 0. This energy clearly loses control on the full gradient of u as ε 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_ε (u_ε) C is compact in W^s, 3₋₎₂ (Ω) for s<1/2. Moreover, this order 1/2 of differentiability is optimal, in the sense that any map u W^1/2, 4 (Ω; S¹) is a limit of a bounded energy sequence. We also establish compactness of boundary traces in L¹ (Ω), and characterize the Γ-limit in the simpler case of maps of a single variable and in the case of a thin-film model.
Bronsard et al. (Wed,) studied this question.