We consider a variational model for heterogeneous phase separation. In particular, we investigate the asymptotic behavior of the first variation $\lambda_\epsilon$ of a heterogeneous variant of the Modica-Mortola energy as the width of the diffuse interface $\varepsilon $ goes to 0. Our convergence result corresponds to a Gibbs-Thomson relation for heterogeneous surface tension.
Starting from this information, one can show that (weak) solutions of the Allen–Cahn equation with spatially dependent double well potential $W$ converge to BV solutions to weighted mean curvature flow (under an energy convergence hypothesis). Additionally, we establish a weak-strong uniqueness principle for BV solutions to weighted mean curvature flow by means of the relative energy technique.