Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Serrin's overdetermined problem on the sphere   We study Serrin's overdetermined boundary value problem \begin{equation*}
  -\Delta_{S^N}\, u=1 \quad \text{ in $\Omega$},\qquad u=0, \; \partial_\eta
u=\textrm{const} \quad \text{on $\partial \Omega$} \end{equation*} in
subdomains $\Omega$ of the round unit sphere $S^N \subset \mathbb{R}^{N+1}$,
where $\Delta_{S^N}$ denotes the Laplace-Beltrami operator on $S^N$. A
subdomain $\Omega$ of $S^N$ is called a Serrin domain if it admits a solution
of this overdetermined problem. In our main result, we construct Serrin domains
in $S^N$, $N \ge 2$ which bifurcate from symmetric straight tubular
neighborhoods of the equator. Our result provides the first example of Serrin
domains in $S^{N}$ which are not bounded by geodesic spheres.

13
The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
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The categories of this paper are: econ
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The categories of this paper are: nlin
22
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: q-alg
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The categories of this paper are: alg-geom