Title: A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime,
Abstract: We consider bounded solutions of the nonlocal Allen-Cahn equation $$
(-\Delta)^s u=u-u^3\qquad{\mbox{ in }}{\mathbb{R}}^3,$$ under the monotonicity
condition $\partial_{x_3}u>0$ and in the genuinely nonlocal regime in
which~$s\in\left(0,\frac12\right)$. Under the limit assumptions $$
\lim_{x_n\to-\infty} u(x',x_n)=-1\quad{\mbox{ and }}\quad \lim_{x_n\to+\infty}
u(x',x_n)=1,$$ it has been recently shown that~$u$ is necessarily $1$D, 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
Ennio De Giorgi.