 Solution

At fixed average order parameter, a nonconvex local potential need not describe the equilibrium free energy. A mixture of two phases can lower it to the common-tangent convex envelope. If their magnetizations are $m_1,m_2$, the coexistence field satisfiesEquivalently, ${\int }_{m_1}^{m_2}[f^{\prime }(m)-h_{\mathrm{coex}}]dm=0$, the equal-area Maxwell construction. An average $\bar{m}$ between $m_1$ and $m_2$ is obtained by choosing their volume fractions according to the lever rule. The gradient term in Landau-Ginzburg theory gives a finite interfacial energy and smooth domain walls; in the thermodynamic limit the bulk gain dominates the interface cost. Domains are therefore especially natural under a constrained mean magnetization or suitable boundary conditions. Without such a constraint, one pure phase is also an equilibrium state at coexistence.