Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 1 / c / Solution

Equilibrium minimizes $F$ at fixed total composition $\int \varphi$, so a Lagrange multiplier gives $\delta F/\delta \varphi ={\mu }_0$, independent of position. In the two homogeneous phases,The symmetric coexistence pair therefore has ${\mu }_0=0$. For a planar profile depending only on the normal coordinate $x$,orup to reversal of the two phases.