闭可平行化流形上具小体积约束的多相 Allen-Cahn-Hilliard 系统解的多重性
Multiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds
AI总结：
本文利用 Lusternik-Schnirelmann 和无穷维 Morse 理论，证明了在闭可平行化黎曼流形上具小体积约束的三相 ACH 矢量方程存在多重解，并给出依赖于流形拓扑不变量的解数量下界，同时证明了解非退化度量集合的通用性。
摘要：
我们证明了在闭可平行化黎曼流形上，具小体积约束和三井（三相）势的 Allen-Cahn-Hilliard (ACH) 矢量方程（含两个方程）存在多重解。更准确地说，我们找到了一个依赖于底层流形某些拓扑不变量的解数量的下界。相变势被认为具有有限的泛极小值集，且在该处取值为零，并在无穷远处具有次临界增长。我们的策略是对矢量能量泛函应用 Lusternik-Schnirelmann 理论和无穷维 Morse 理论。为此，我们利用了相关的 ACH 能量 $Γ$-收敛于簇的加权多周长这一性质，结合等周理论中的一些深刻定理，为应用摄影法提供了合适的框架。在此过程中，簇的多等周函数缺乏闭解析表达式带来了一个棘手的问题。此外，利用横截性定理，我们还证明了使得 ACH 系统解非退化的度量集合的通用性。
英文摘要：
We prove the existence of multiple solutions to the Allen--Cahn--Hilliard (ACH) vectorial equation (with two equations) involving a triple-well (triphasic) potential with a small volume constraint on a closed parallelizable Riemannian manifold. More precisely, we find a lower bound for the number of solutions depending on some topological invariants of the underlying manifold. The phase transition potential is considered to have a finite set of global minima, where it also vanishes, and a subcritical growth at infinity. Our strategy is to employ the Lusternik--Schnirelmann and infinite-dimensional Morse theories for the vectorial energy functional. To this end, we exploit that the associated ACH energy $Γ$-converges to the weighted multi-perimeter for clusters, which combined with some deep theorems from isoperimetric theory yields the suitable setup to apply the photography method. Along the way, the lack of a closed analytic expression for the multi-isoperimetric function for clusters imposes a delicate issue. Furthermore, using a transversality theorem, we also show the genericity of the set of metrics for which solutions to the ACH system are nondegenerate.
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