A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime

We consider bounded solutions of the nonlocal Allen–Cahn equation

$$\begin{aligned} (-\Delta )^s u=u-u^3\qquad { \text{ in } }\mathbb {R}^3, \end{aligned}$$

under the monotonicity condition \(\partial _{x_3}u>0\) and in the genuinely nonlocal regime in which \(s\in \left( 0,\frac{1}{2}\right) \). Under the limit assumptions

$$\begin{aligned} \lim _{x_n\rightarrow -\infty } u(x',x_n)=-1\quad { \text{ and } }\quad \lim _{x_n\rightarrow +\infty } u(x',x_n)=1, \end{aligned}$$

it has been recently shown in Dipierro et al. (Improvement of flatness for nonlocal phase transitions, 2016) that u is necessarily 1D, i.e. it depends only on one Euclidean variable. The goal of this paper is to obtain a similar result without assuming such limit conditions. This type of results can be seen as nonlocal counterparts of the celebrated conjecture formulated by De Giorgi (Proceedings of the international meeting on recent methods in nonlinear analysis (Rome, 1978), Pitagora, Bologna, pp 131–188, 1979).