Abstract
We prove the existence of nontrivial global minimizers of the Allen–Cahn equation in dimension 8 and above. More precisely, given any strict area-minimizing Lawson's cone, there is a family of global minimizers whose nodal sets are asymptotic to this cone. As a consequence of Jerison–Monneau's program we then establish the existence of many new counter-examples to the De Giorgi conjecture whose nodal sets are different from the Bombieri–De Giorgi–Giusti minimal graph.
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