Half-space theorems for the Allen-Cahn equation and related problems
Résumé
In this paper we obtain rigidity results for a bounded non-constant entire solution u of the Allen-Cahn equation in R n , whose level set {u = 0} is contained in a half-space. If n ≤ 3 we prove that the solution must be one-dimensional. In dimension n ≥ 4, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.
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