论文标题
Allen-Cahn方程的马鞍解决方案的稳定性
Stability of the saddle solutions for the Allen-Cahn equation
论文作者
论文摘要
我们关注由Cabré和Terra \ cite {C，C2}构建的Allen-CaHn方程的鞍座解决方案，in $ \ Mathbb {r}^{2M}^{2M}％= \ Mathbb {r}^{M}^{M}^{M}^{M} \ times \ times \ MathBB {r}^r}^^^m}^{m} $。这些解决方案精确地消失在西蒙斯锥体上。 \ cite {c，c2，c1}中显示了马鞍解的存在和独特性。关于稳定性，schatzman \ cite {sch}证明，萨德尔解决方案对于$ m = 1，$cabré\ cite {c1}表明了$ m = 2,3 $的不稳定，而$ m \ geq7 $的稳定性。这已经打开了$ M = 4,5,6 $的情况。在本文中，我们表明，当$ M = 4,5,6 $时，马鞍解决方案是稳定的，从而证实了Cabré在\ cite {c1}中的猜想。在维度上的马鞍解决方案$ 200M \ geq8 $的猜想应该是能量功能的全球最小化器。
We are concerned with the saddle solutions of the Allen-Cahn equation constructed by Cabré and Terra \cite{C,C2} in $\mathbb{R}^{2m}% =\mathbb{R}^{m}\times\mathbb{R}^{m}$. These solutions vanish precisely on the Simons cone. The existence and uniqueness of saddle solution are shown in \cite{C,C2,C1}. Regarding the stability, Schatzman \cite{Sch} proved that the saddle solution is unstable for $m=1,$ Cabré \cite{C1} showed the instability for $m=2,3$ and stability for $m\geq7$. This has left open the case of $m=4,5,6$. In this paper we show that the saddle solutions are stable when $m=4,5,6$, thereby confirming Cabré's conjecture in \cite{C1}. The conjecture that saddle solutions in dimensions $2m\geq8$ should be global minimizers of the energy functional remains open.
