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.

翻译：利用 $\Gamma$-收敛方法，我们研究界面宽度参数 $\varepsilon > 0$ 的 Cahn-Hilliard 问题在带锥奇点流形上的相变现象。我们证明相应能量泛函的极小元存在，且当 $\varepsilon \to 0$ 时收敛到一个仅取两个值的函数，其界面沿满足体积约束的极小超曲面。在数值算例中，我们采用延拓与分岔方法研究二维椭圆锥上小参数 $\varepsilon > 0$ 时临界点族，该族由锥高与底面椭圆度参数化。其中部分临界点为界面穿过锥尖的极小元。另一方面，我们证明对于一般锥角小于 $2\pi$ 的锥，对应 $\varepsilon = 0$ 的周长泛函极小界面永远不会穿过锥尖。因此有限 $\varepsilon > 0$ 时的锥尖极小元在 $\varepsilon \to 0$ 时必须变为鞍点，并通过数值方法识别出相关分岔，在示例中发现了 $\varepsilon > 0$ 与锥参数之间的微妙相互作用。