We prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations that includes $(-\Delta)^{s/2} u = u-u^3$, with~$s\in(0,1)$, we obtain a result in the same spirit of a celebrated theorem of Savin for the equation $-\Delta u = u-u^3$. As a consequence, we deduce that entire solutions to~$(-\Delta)^{s/2} u = u-u^3$ with asymptotically flat level sets are $1$D when~$s\in(0,1)$. The results presented are completely new even for the case of the fractional Laplacian, but the robustness of the proofs allows us to treat also more general, possibly anisotropic, integro-differential operators.

中文翻译：

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非局部相变平坦度的改进

我们证明了非局部相变的平坦度结果的改进。对于一类包含 $(-\Delta)^{s/2} u = uu^3$, with ~$s\in(0,1)$ 的非局部方程，我们得到了与 a方程 $-\Delta u = uu^3$ 的著名 Savin 定理。因此，我们推导出~$(-\Delta)^{s/2} u = uu^3$ 与渐近平坦水平集的整个解是 $1$D 当~$s\in(0,1)$ . 即使对于分数拉普拉斯算子的情况，所呈现的结果也是全新的，但证明的稳健性使我们能够处理更一般的，可能是各向异性的，积分微分算子。