For all n≥1, we are interested in bounded solutions of the Allen-Cahn equation δu+u-u3=0 which are defined in all Rn+1 and whose zero set is asymptotic to a given minimal cone. In particular, in dimension n+1≥8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes. © 2012.
Pacard, F., & Wei, J. (2013). Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones. Journal of Functional Analysis, 264(5), 1131–1167. https://doi.org/10.1016/j.jfa.2012.03.010