A weighted gradient theory of phase transitions with a possibly singular and degenerate spatial inhomogeneity

This paper studies the asymptotic behavior of a perturbed variational problem for the Cahn–Hilliard theory of phase transitions in a fluid, with spatial inhomogeneities in the internal free energy term. The inhomogeneous term can vanish or become infinite and it can also behave as an appropriate power of the distance from the boundary. The standard minimal interface criterion will be recovered even in spite of such severe degeneracies and/or singularities.