Half space theorem for the Allen-Cahn equation and related problems
| Autor: | Hamel, Francois, Liu, Yong, Sicbaldi, Pieralberto, Wang, Kelei, Wei, Juncheng |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In this paper we obtain rigidity results for a bounded non-constant entire solution $u$ of the Allen-Cahn equation in $\mathbb{R}^n$, whose level set $\{u=0\}$ is contained in a half-space. If $n\leq 3$ we prove that the solution must be one-dimensional. In dimension $n\geq 4$, 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. Comment: 19 pages; replacing version two |
| Databáze: | arXiv |
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