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.

中文翻译：

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具有圆锥奇点的流形上的相变和最小界面

使用 $\Gamma$ 收敛性，我们研究了具有圆锥奇点的流形上的相变的界面宽度参数 $\varepsilon > 0$ 的 Cahn-Hilliard 问题。我们证明了相应能量函数的极小值存在并收敛为一个函数，如 $\varepsilon \to 0$，该函数仅采用两个值，其界面沿超曲面，该超曲面在满足体积约束的曲面中具有最小面积。在数值示例中，我们使用延拓和分岔方法来研究二维椭圆锥上小 $\varepsilon > 0$ 处的临界点族，通过底面的高度和椭圆度进行参数化。其中一些关键点是具有穿过锥尖的界面的最小化器。另一方面，我们证明，对于角度小于 $2\pi$ 的一般圆锥来说，对应于 $\varepsilon = 0$ 的周长泛函的最小化界面永远不会穿过圆锥尖端。因此，有限 $\varepsilon > 0$ 的尖端最小化器必须成为鞍点，因为 $\varepsilon \to 0$，并且我们以数字方式识别相关的分叉，在我们的示例中找到 $\varepsilon > 0$ 和圆锥参数的微妙相互作用。