Phase transitions and minimal interfaces on manifolds with conical singularities

Using $\Gamma$-convergence, we study the Cahn-Hilliard problem with interface width parameter $\varepsilon > 0$ for phase transitions on manifolds with conical singularities. We prove that minimizers of the corresponding energy functional exist and converge, as $\varepsilon \to 0$, to a function that takes only two values with an interface along a hypersurface that has minimal area among those satisfying a volume constraint. In a numerical example, we use continuation and bifurcation methods to study families of critical points at small $\varepsilon > 0$ on 2D elliptical cones, parameterized by height and ellipticity of the base. Some of these critical points are minimizers with interfaces crossing the cone tip. On the other hand, we prove that interfaces which are minimizers of the perimeter functional, corresponding to $\varepsilon = 0$, never pass through the cone tip for general cones with angle less than $2\pi$. Thus tip minimizers for finite $\varepsilon > 0$ must become saddles as $\varepsilon \to 0$, and we numerically identify the associated bifurcation, finding a delicate interplay of $\varepsilon > 0$ and the cone parameters in our example.