双稳态扩散方程鞍形解的定性性质
Qualitative properties of saddle-shaped solutions to bistable diffusion equations
摘要：
在整个$\R^{2m}$中考虑椭圆型方程$-Δu = f(u)$，其中$f$是双稳态.已知在$\R^{2m}$中存在鞍形解。这是一个在$\R^{2m}$中改变符号并且只在Simons锥${\mathcal C}=\{(x^1,x^2)\in\R^m\times\R^m: |x^1|=|x^2|\}$上消失的解。众所周知，这些解在2维和4维中是不稳定的。本文证明了当$2m=6$时，每个鞍形解在每个紧集外都是不稳定的，从而具有无穷Morse指数.为此，我们建立了鞍形解在无穷远处的渐近行为。此外，我们还证明了一个最小和一个最大鞍形解的存在性，并得到了最大解的单调性。这些结果是有关的猜想德乔治一维对称性的某些解决方案。鞍形的解决方案是最简单的候选人，除了一维的解决方案，是全球最小的高维，尚未建立的属性。
英文摘要：
We consider the elliptic equation $-Δu = f(u)$ in the whole $\R^{2m}$, where $f$ is of bistable type. It is known that there exists a saddle-shaped solution in $\R^{2m}$. This is a solution which changes sign in $\R^{2m}$ and vanishes only on the Simons cone ${\mathcal C}=\{(x^1,x^2)\in\R^m\times\R^m: |x^1|=|x^2|\}$. It is also known that these solutions are unstable in dimensions 2 and 4. In this article we establish that when $2m=6$ every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution. These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.
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