Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation

作者：Cabre Xavier^*

来源：Journal de Mathematiques Pures et Appliquees, 2012, 98(3): 239-256.

DOI：10.1016/j.matpur.2012.02.006

We establish the uniqueness of a saddle-shaped solution to the diffusion equation -Delta u = f (u) in all of R-2m, where f is of bistable type, in every even dimension 2m %26gt;= 2. In addition, we prove its stability whenever 2(m) %26gt;= 14. %26lt;br%26gt;Saddle-shaped solutions are odd with respect to the Simons cone C = {(x(1), x(2)) is an element of R-m x R-m: vertical bar x(1)vertical bar = vertical bar x(2)vertical bar} 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 %26gt;= 8, saddle-shaped solutions are expected to be global minimizers when 2m %26gt;= 8, or at least in higher dimensions. This is a property stronger than stability which is not yet established in any dimension.

• 出版日期2012-9