Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A right inverse of the divergence for planar H\"older-$\alpha$ domains   If $\Omega\subset\R^n$ is a bounded domain, the existence of solutions ${\bf
u}\in H^1_0(\Omega)^n$ of ${div} {\bf u} = f$ for $f\in L^2(\Omega)$ with
vanishing mean value, is a basic result in the analysis of the Stokes
equations.
  In particular it allows to show the existence of a solution $({\bf u},p)\in
H^1_0(\Omega)^n\times L^2(\Omega)$, where ${\bf u}$ is the velocity and $p$ the
pressure.
  It is known that the above mentioned result holds when $\Omega$ is a
Lipschitz domain and that it is not valid for arbitrary H\"older-$\alpha$
domains.
  In this paper we prove that if $\Omega$ is a planar simply connected
H\"older-$\alpha$ domain, there exist right inverses of the divergence which
are continuous in appropriate weighted spaces, where the weights are powers of
the distance to the boundary. Moreover, we show that the powers of the distance
in the results obtained are optimal.
  In our results, the zero boundary condition is replaced by a weaker one. For
the particular case of domains with an external cusp of power type, we prove
that our weaker boundary condition is equivalent to the standard one. In this
case we show the well posedness of the Stokes equations in appropriate weighted
Sobolev spaces obtaining as a consequence the existence of a solution $({\bf
u},p)\in H^1_0(\Omega)^n\times L^r(\Omega)$ for some $r<2$ depending on the
power of the cusp.

20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn