\(\mathbb{R}^n\) 中具有 \((n + 1)\) 个结的向量 Allen-Cahn 的局部极小值
Local minimizers in $\mathbb{R}^n$ of vector Allen-Cahn with an $(n+1)$-junction
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中文总结 AI 辅助
研究 \(\mathbb{R}^n\) 中单位球变形区域 \(\Omega\) 上向量 Allen-Cahn 能量的局部极小值，通过证明划分是加权周长问题的孤立局部极小值，建立其存在性及收敛性，推广了 \(n = 3\) 时结果并弱化关键假设。
中文摘要 AI 辅助
对于作为 \(\mathbb{R}^n\) 中单位球变形的区域 \(\Omega\)，我们建立了具有 \(n + 1\) 个阱的向量 Allen-Cahn 能量的一系列局部极小值的存在性。该序列在 \(L^1\) 拓扑中收敛到 \(\Omega\) 的一个划分，其骨架由包含一个 \((n + 1)\) 结的单纯形锥给出。这是通过证明该划分是作为 Allen-Cahn 泛函序列的相关 \(\Gamma -\) 极限出现的加权周长问题的孤立局部极小值来实现的。本文的结果推广了作者与 Peter Sternberg 早期文章（MR5033050）中的结果，早期文章处理的是 \(n = 3\) 的情况。我们还弱化了作者与 Peter Sternberg 早期文章中的一个关键假设。
英文摘要
For a domain $Ω$ that is a deformation of a unit ball in $\mathbb{R}^n$, we establish the existence of a sequence of local minimizers for the vector Allen-Cahn energy having $n+1$ wells. This sequence converges in the $L^1$ topology to a partition of $Ω$ whose skeleton is given by a simplex cone that contains an $(n+1)$-junction point. This is accomplished by proving that the partition is an isolated local minimizer of a weighted perimeter problem arising as the associated $Γ$-limit of the sequence of Allen-Cahn functionals. The results established in this article generalize those in the author's earlier article with Peter Sternberg (MR5033050), which dealt with the case $n=3$. We also weaken the one crucial assumption from the author's earlier article with Peter Sternberg (MR5033050).
