Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: One-dimensional symmetry for integral systems in two dimensions   The purpose of this brief paper is to prove De Giorgi type results for stable
solutions of the following nonlocal system of integral equations in two
dimensions $$ L(u_i) = H_i(u) \quad \text{in} \ \ \mathbb R^2 , $$ where
$u=(u_i)_{i=1}^m$ for $u_i: \mathbb R^n\to \mathbb R$, $H=(H_i)_{i=1}^m$ is a
general nonlinearity. The operator $L$ is given by $$L(u_i (x)):= \int_{\mathbb
R^2} [u_i(x) - u_i(z)] K(z-x) dz,$$ for some kernel $K$. The idea is to apply a
linear Liouville theorem for the quotient of partial derivatives, just like in
the proof of the classical De Giorgi's conjecture in lower dimensions. Since
there is no Caffarelli-Silvestre local extension problem associated to the
above operator, we deal with this problem directly via certain integral
estimates.

1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
19
The categories of this paper are: eess
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th