Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local analysis of a two phase free boundary problem concerning mean
  curvature   We consider an overdetermined problem for a two phase elliptic operator in
divergence form with piecewise constant coefficients. We look for domains such
that the solution $u$ of a Dirichlet boundary value problem also satisfies the
additional property that its normal derivative $\partial_n u$ is a multiple of
the radius of curvature at each point on the boundary. When the coefficients
satisfy some "non-criticality" condition, we construct nontrivial solutions to
this overdetermined problem employing a perturbation argument relying on shape
derivatives and the implicit function theorem. Moreover, in the critical case,
we employ the use of the Crandall-Rabinowitz theorem to show the existence of a
branch of symmetry breaking solutions bifurcating from trivial ones. Finally,
some remarks on the one phase case and a similar overdetermined problem of
Serrin type are given.

16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph