Partial Sobolev spaces and anisotropic smectic liquid crystals

The partial Sobolev spaces with respect to a vector field are introduced, and are used to study minimization problems of the functionals which are degenerate in the sense that they do not have control on either the tangential part or the perpendicular part of the magnetic gradients. Based on these results we obtain the asymptotic behavior of the minimizers of the anisotropic Landau-de Gennes functional of smectic liquid crystals, as one of the anisotropy coefficients approaches to zero.