We study solutions of Allen-Cahn equation:

$$-\Delta u=u-u^3 \text{ in } \mathbb{R}^n,$$

corresponding to the energy functional

$$J(u)=\int |\nabla u|^2+\frac{1}{2}(u^2-1)^2.$$

%--- Abstract

We show there is a family of global minimizers of this functional in dimension 8. The main ingredient used is the family of foliated solutions constructed by Pacard-Wei. This together with a result of Jerison-Monneau give us new counter-examples of the De Giorgi conjecture in dimension 9. We also discuss related questions in a free boundary problem. This is joint work with Kelei Wang and Juncheng Wei.

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%\begin{thebibliography}{99}

%\bibitem{bib1}

%\newblock Au. Thor,

%\newblock \emph{The Most Important Equations of the Universe,}

%\newblock A Noble Publishing Co, Pleasant City, 1969.

%\end{thebibliography}

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