Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solutions of some Monge-Amp\`ere equations with isolated and line
  singularities   In this paper, we study existence, regularity, classification, and
asymptotical behaviors of solutions of some Monge-Amp\`ere equations with
isolated and line singularities. We classify all solutions of $\det \nabla^2
u=1$ in $\R^n$ with one puncture point. This can be applied to characterize
ellipsoids, in the same spirit of Serrin's overdetermined problem for the
Laplace operator. In the case of having $k$ non-removable singular points for
$k>1$, modulo affine equivalence the set of all generalized solutions can be
identified as an explicit orbifold of finite dimension. We also establish
existence of global solutions with general singular sets, regularity
properties, and optimal estimates of the second order derivatives of
generalized solutions near the singularity consisting of a point or a straight
line. The geometric motivation comes from singular semi-flat Calabi-Yau
metrics.

15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom