Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two dimensional liquid crystal droplet problem with tangential boundary
  condition   This paper studies a shape optimization problem which reduces to a nonlocal
free boundary problem involving perimeter. It is motivated by a study of liquid
crystal droplets with a tangential anchoring boundary condition and a volume
constraint. We establish in 2D the existence of an optimal shape that has two
cusps on the boundary. We also prove the boundary of the droplet is a chord-arc
curve with its normal vector field in the VMO space. In fact, the boundary
curves of such droplets belong to the so-called Weil-Petersson class. In
addition, the asymptotic behavior of the optimal shape when the volume becomes
extremely large or small is also studied.

14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
18
The categories of this paper are: econ
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph