Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Linearity of homogeneous solutions to degenerate elliptic equations in
  dimension three   Given a linear elliptic equation $\sum a_{ij} u_{ij} =0$ in $\mathbb{R}^3$,
it is a classical problem to determine if its degree-one homogeneous solutions
$u$ are linear. The answer is negative in general, by a construction of
Martinez-Maure. In contrast, the answer is affirmative in the uniformly
elliptic case, by a theorem of Han, Nadirashvili and Yuan, and it is a known
open problem to determine the degenerate ellipticity condition on $(a_{ij})$
under which this theorem still holds. In this paper we solve this problem. We
prove the linearity of $u$ under the following degenerate ellipticity condition
for $(a_{ij})$, which is sharp by Martinez-Maure example: if $\mathcal{K}$
denotes the ratio between the largest and smallest eigenvalues of $(a_{ij})$,
we assume $\mathcal{K}|_{\mathcal{O}}$ lies in $L_{\rm loc}^1$ for some
connected open set $\mathcal{O}\subset \mathbb{S}^2$ that intersects any
configuration of four disjoint closed geodesic arcs of length $\pi$ in
$\mathbb{S}^2$. Our results also give the sharpest possible version under which
an old conjecture by Alexandrov, Koutroufiotis and Nirenberg (disproved by
Martinez-Maure's example) holds.

28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex