论文标题
对多最多的Allen-Cahn-Hilliard系统的多种解决方案，对封闭的可行歧管有少量约束
Multiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds
论文作者
论文摘要
我们证明了艾伦（Allen）的多种溶液 - cahn-hilliard（ach）矢量方程（具有两个方程式），涉及三孔（三孔）电位，并且在封闭的可行的Riemannian歧管上具有较小的体积约束。更确切地说，根据基础歧管的某些拓扑不变，我们发现了解决方案数量的下限。相转换电位被认为具有有限的全球最小值，它也会消失，而无限属性则具有亚临界增长。我们的策略是利用Lusternik-Schnirelmann和无限二维摩尔斯科理论来效力。 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.在此过程中，缺乏用于集群的多相位功能的封闭分析表达会引发一个微妙的问题。此外，使用横向定理，我们还显示了对ACH系统解决方案的一组指标的通用性。
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.
