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Our main results include: (1) obtaining a complete phase diagram with explicit analytic formulae of Thomas--Fermi solutions on the parameter $q$-$M$ plane ($q$: quadratic Zeeman energy, $M$: total magnetization)  for both ferromagnetic and antiferromagnetic systems, and (2) an entire characterization of ground-state patterns in the Thomas--Fermi regime and the semi-classical regime.  In particular, a class of interesting ground states are mixed states. A mixed state consists of two constant states separated by an interface.

This interface  minimizes an effective energy,  the sum of an interfacial energy and two boundary energies.  As a result, the interface has constant mean curvature and the corresponding contact angle satisfies Young's relation as that in the classical wetting process.

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