Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones

摘要

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.