Uniqueness and stability of saddle-shaped solutions to the Allen–Cahn equation
Uniqueness and stability of saddle-shaped solutions to the Allen–Cahn equation
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DOI：
10.1016/j.matpur.2012.02.006
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发表时间：
2011-02
期刊：
影响因子：
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通讯作者：
X. Cabré
中科院分区：
文献类型：
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作者：
X. Cabré
我们在所有 R2m 中建立了扩散方程 −Δu=f(u) 的鞍形解的唯一性，其中 f 是双稳态类型，在每个偶数维度 2m⩾2 中。此外，我们证明了每当 2m⩾14 时它的稳定性。鞍形解相对于西蒙斯锥 C={(x1,x2)∈Rm×Rm:|x1|=|x2|} 是奇数，并且存在于所有偶数维度中。只有当 2m=2 时才知道它们的独特性。另一方面，已知它们在 2、4 和 6 维上不稳定。它们在 8、10 和 12 维上的稳定性仍然是一个悬而未决的问题。此外，由于西蒙斯锥在 2m⩾8 时使面积最小化，因此鞍形解决方案预计在 2m⩾8 时或至少在更高维度上成为全局最小化器。这是一种比稳定性更强的特性，但在任何维度上都尚未建立。
We establish the uniqueness of a saddle-shaped solution to the diffusion equation −Δu=f(u) in all of R2m, where f is of bistable type, in every even dimension 2m⩾2. In addition, we prove its stability whenever 2m⩾14. Saddle-shaped solutions are odd with respect to the Simons cone C={(x1,x2)∈Rm×Rm:|x1|=|x2|} and exist in all even dimensions. Their uniqueness was only known when 2m=2. On the other hand, they are known to be unstable in dimensions 2, 4, and 6. Their stability in dimensions 8, 10, and 12 remains an open question. In addition, since the Simons cone minimizes area when 2m⩾8, saddle-shaped solutions are expected to be global minimizers when 2m⩾8, or at least in higher dimensions. This is a property stronger than stability which is not yet established in any dimension.