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

By Frank Pacard and Juncheng Wei


For all n ≥ 1, we are interested in bounded solutions of the Allen-Cahn equation ∆u + u − u 3 = 0 which are defined in all R n+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

Year: 2011
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