Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a Bernoulli-type overdetermined free boundary problem   In this article we study a Bernoulli-type free boundary problem and
generalize a work of Henrot and Shahgholian in \cite{HS1} to
$\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose
structure is modeled on the $p$-Laplace equation for a fixed $1<p<\infty$. In
particular, we show that if $K$ is a bounded convex set satisfying the interior
ball condition and $c>0$ is a given constant, then there exists a unique convex
domain $\Omega$ with $K\subset \Omega$ and a function $u$ which is
$\mathcal{A}$-harmonic in $\Omega\setminus K$, has continuous boundary values
$1$ on $\partial K$ and $0$ on $\partial\Omega$, such that $|\nabla u|=c$ on
$\partial \Omega$. Moreover, $\partial\Omega$ is $C^{1,\gamma}$ for some
$\gamma>0$, and it is smooth provided $\mathcal{A}$ is smooth in $\mathbb{R}^n
\setminus \{0\}$. We also show that the super level sets $\{u>t\}$ are convex
for $t\in (0,1)$.

18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc