Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $$
(-\Delta)^s u=u-u^3\qquad{\mbox{ in }}{\mathbb{R}}^3,$$ 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.

19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
14
The categories of this paper are: math-ph