Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: One-phase free-boundary problems with degeneracy   In this paper, we study local minimizers of a degenerate version of the
Alt-Caffarelli functional. Specifically, we consider local minimizers of the
functional $J_{Q}(u, \Omega):= \int_{\Omega} |\nabla u|^2 +
Q(x)^2\chi_{\{u>0\}}dx$ where $Q(x) = dist(x, \Gamma)^{\gamma}$ for $\gamma>0$
and $\Gamma$ a $C^{1, \alpha}$ submanifold of dimension $0 \le k \le n-1$. We
show that the free boundary may be decomposed into a rectifiable set, on which
we prove effective estimates, and a degenerate cusp set, about which little can
be said in general with the current techniques. Work in the theory of water
waves and the Stokes wave serves both as our inspiration and as an application.
However, the main thrust of this paper is to inaugurate a study of the geometry
of the free boundary for degenerate one-phase Bernoulli free-boundary problems.

13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th