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The saddle-shaped solution to the Allen-Cahn equation and a conjecture of De Giorgi

Banff International Research Station for Mathematical Innovation and Discovery

I will discuss some questions regarding the conjecture of De Giorgi on the Allen-Cahn equation and which remain still open. The talk will be mainly concerned with the saddle-shaped solution in all of $\R^{2m}$. A remarkable open problem is to establish that this solution is a minimizer in high dimensions ---more precisely, this is believed to be true for $2m \geq 8$. The saddle-shaped solution is odd with respect to the Simons cone and exists in all even dimensions. I will explain results of the author and collaborators which establish: the uniqueness of the saddle-shaped in every even dimension $2m \geq 2$, its instability in dimensions 2, 4, and 6, and its stability for $2m \geq 14$. I will also describe results of Pacard and Wei, and a very recent one by Liu, Wang, and Wei, which construct a family of global minimizers in $\R^8$. If this family includes the saddle-shaped solution is still unknown.

Mathematics; Partial differential equations; Integral equations

video/mp4

Author affiliation: ICREA and Universitat Politecnica de Catalunya

2017-10-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63194

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/