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- W2950595936 abstract "We establish the uniqueness of a saddle-shaped solution to the diffusion equation $-Delta u = f(u)$ in all of $mathbb{R}^{2m}$, where $f$ is of bistable type, in every even dimension $2m geq 2$. In addition, we prove its stability whenever $2m geq 14$. Saddle-shaped solutions are odd with respect to the Simons cone ${mathcal C} = {(x^1,x^2) in mathbb{R}^m times mathbb{R}^m : |x^1|=|x^2| }$ 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 $2m geq 8$, saddle-shaped solutions are expected to be global minimizers when $2m geq 8$, or at least in higher dimensions. This is a property stronger than stability which is not yet established in any dimension." @default.
- W2950595936 created "2019-06-27" @default.
- W2950595936 creator A5035208021 @default.
- W2950595936 date "2011-02-15" @default.
- W2950595936 modified "2023-10-06" @default.
- W2950595936 title "Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation" @default.
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