Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Symmetry results for fractional elliptic systems and related problems   We study elliptic gradient systems with fractional laplacian operators on the
whole space $$ (- \Delta)^\mathbf s \mathbf u =\nabla H (\mathbf u) \ \
\text{in}\ \ \mathbf{R}^n,$$ where $\mathbf u:\mathbf{R}^n\to \mathbf{R}^m$,
$H\in C^{2,\gamma}(\mathbf{R}^m)$ for $\gamma > \max(0,1-2\min \left \{s_i
\right \})$, $\mathbf s=(s_1,\cdots,s_m)$ for $0<s_i<1$ and $\nabla H (\mathbf
u)=(H_{u_i}(u_1, u_2,\cdots,u_m))_{i}$. We prove De Giorgi type results for
this system for certain values of $\mathbf s$ and in lower dimensions, i.e.
$n=2,3$. Just like the local case, the concepts of orientable systems and
$H-$monotone solutions, established in [18], play the key role in proving
symmetry results. In addition, we provide optimal energy estimates, a
monotonicity formula, a Hamiltonian identity and various Liouville theorems.

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The categories of this paper are: physics
25
The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: chem-ph