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Some energy estimates for stable solutions to fractional Allen–Cahn equations

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Abstract

In this paper we study stable solutions to the fractional equation

$$\begin{aligned} (-\Delta )^s u =f(u), \quad |u| < 1 \quad \hbox {in }\mathbb {R}^d, \end{aligned}$$
(0.1)

where \(0<s<1\) and \(f:[-1,1] \rightarrow \mathbb {R}\) is a \(C^{1,\alpha }\) function for \(\alpha >\max \{0, 1-2s\}\). We obtain sharp energy estimates for \(0<s<1/2\) and rough energy estimates for \(1/2 \le s <1\). These lead to a different proof from literature of the fact that when \(d=2, \, 0<s<1\), entire stable solutions to (0.1) are 1-D solutions. The scheme used in this paper is inspired by Cinti–Serra–Valdinoci [16] which deals with stable nonlocal sets, and Figalli–Serra [26] which studies stable solutions to (0.1) for the case \(s=1/2\).

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Notes

  1. 1.

    This condition can be replaced by the more general condition that the limits of u when \(x_d \rightarrow \pm \infty \) are 2D , see [18, Theorem 8.1]

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Acknowledgements

This research is partially supported by NSF Grants DMS-1601885 and DMS-1901914.

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Correspondence to Changfeng Gui.

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Communicated by X. Cabre.

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Gui, C., Li, Q. Some energy estimates for stable solutions to fractional Allen–Cahn equations. Calc. Var. 59, 49 (2020). https://doi.org/10.1007/s00526-020-1701-2

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Mathematics Subject Classification

  • Primary 35B06
  • 35J15
  • 35J20
  • 35J91
  • 53A10