= Solution
The <Maxwell construction> replaces a nonconvex homogeneous <free-energy density> by its coexistence envelope. If $U_0(M)$ excludes the source, the coexisting values $M_1,M_2$ satisfy
$$
U_0'(M_1)=U_0'(M_2)=h_*,\qquad
U_0(M_2)-U_0(M_1)=h_*(M_2-M_1).
$$
Thus one straight line is tangent at both endpoints. Equivalently,
$$
\int_{M_1}^{M_2}[U_0'(M)-h_*]dM=0,
$$
the equal-area form of the <common-tangent construction>. At fixed average $\overline M$ between the endpoints, the <lever rule> gives $\overline M=pM_1+(1-p)M_2$. Such a mixture has the tangent-line bulk energy, below the energy of an unstable homogeneous state.
The two phases occupy spatial domains separated by <domain walls>. The gradient term gives a positive wall energy, but a macroscopic interface has area of order $L^{D-1}$ rather than bulk volume $L^D$, so its cost per volume vanishes in the <thermodynamic limit>. For the symmetric quartic example a <scalar quartic domain wall> obeys $\kappa\phi''=r\phi+u\phi^3$ and interpolates between $\pm M_0$ as $\phi=M_0\tanh(x/\ell)$, with $M_0^2=|r|/u$ and $\ell=\sqrt{2\kappa/|r|}$. Multiplication of the differential equation by $\phi'$ gives the first integral $\kappa(\phi')^2/2=u(\phi^2-M_0^2)^2/4$, explaining the finite positive interface cost. A uniform unconstrained system need not form both domains: phase separation is particularly relevant when the average <order parameter> is fixed or boundaries enforce competing phases.
<Hysteresis> occurs when the system follows a <metastable> local minimum after another minimum has become the equilibrium one. For $r<0,u>0$, the equation of state is $h=rM+uM^3$. The <hysteresis spinodals of a scalar quartic potential> follow from $r+3uM^2=0$:
$$
\boxed{M_{\rm sp}=\pm\sqrt{\frac{|r|}{3u}},\qquad |h_{\rm sp}|=\frac{2|r|^{3/2}}{3\sqrt{3u}}.}
$$
Following local minima until these limits gives different switching fields on increasing and decreasing $h$. Equilibrium instead switches at $h=0$. Actual switching can occur before a mean-field spinodal through nucleation, and depends on time scales; hysteresis is not a property of a globally minimized equilibrium potential. The second panel below distinguishes the local branches and their limiting switches.
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