Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sobolev inequalities for the Hardy-Schr\"odinger operator: Extremals and
  critical dimensions   In this expository paper, we consider the Hardy-Schr\"odinger operator
$-\Delta -\gamma/|x|^2$ on a smooth domain \Omega of R^n with 0\in\bar{\Omega},
and describe how the location of the singularity 0, be it in the interior of
\Omega or on its boundary, affects its analytical properties. We compare the
two settings by considering the optimal Hardy, Sobolev, and the
Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites:
$C(\int_{\Omega}\frac{u^{p}}{|x|^s}dx)^{\frac{2}{p}}\leq \int_{\Omega} |\nabla
u|^2dx-\gamma \int_{\Omega}\frac{u^2}{|x|^2}dx$ for all $u\in H^1_0(\Omega)$,
where \gamma <n^2/4, s\in [0,2) and p:=2(n-s)/(n-2). We address questions
regarding the explicit values of the optimal constant C, as well as the
existence of non-trivial extremals attached to these inequalities. Scale
invariance properties lead to situations where the best constants do not depend
on the domain and are not attainable. We consider two different approaches to
"break the homogeneity" of the problem:
  One approach was initiated by Brezis-Nirenberg and by Janelli. It is suitable
for the case where 0 is in the interior of \Omega, and consists of considering
lower order perturbations of the critical nonlinearity. The other approach was
initiated by Ghoussoub-Kang , C.S. Lin et al. and Ghoussoub-Robert. It consists
of considering domains where the singularity is on the boundary.
  Both of these approaches are rich in structure and in challenging problems.
If 0\in \Omega, a negative linear perturbation suffices for higher dimensions,
while a positive "Hardy-singular interior mass" is required in lower
dimensions. If the singularity is on the boundary, then the local geometry
around 0 plays a crucial role in high dimensions, while a positive
"Hardy-singular boundary mass" is needed for the lower dimensions.

0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys