  We develop an analytic theory for the ground state patterns and their phase
transitions for spin-1 Bose-Einstein condensates on a bounded domain in the
presence of a uniform magnetic field. Within the Thomas-Fermi approximation,
these ground state patterns are composed of four basic states: magnetic state,
nematic state, two-component state and three-component state, separated by
interfaces. A complete phase diagram of the ground state patterns are found
analytically with different quadratic Zeeman energy q and total magnetization M
for both ferromagnetic and antiferromagnetic systems. Using the
\Gamma-convergence technique, it is found that the semi-classical limits of
these ground states minimize an energy functional which consists of interior
interface energy plus a boundary contact energy. As a consequence, the
interface between two different basic states has constant mean curvature, and
the contact angle between the interface and the boundary obeys Young's
relation.
