We deal with reaction-diffusion equations of bistable type in an inhomogeneous
medium. When the reaction term is balanced in the sense that a bulk potential
energy attains the same global minimum at the two stable equilibria for each spatial
point, we derive a free-boundary problem whose solutions determine equilibirum interfaces.
We show that a non-degenerate solution of the free-boundary problem gives
rise to an equilibrium internal layer solution of the reaction-diffusion equation, and
moreover, the stability property of the latter is obtained from a linearization of the free
boundary problem.