Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regularity of solutions to fully nonlinear elliptic and parabolic free
  boundary problems   We consider fully nonlinear obstacle-type problems of the form
\begin{equation*} \begin{cases} F(D^{2}u,x)=f(x) & \text{a.e.
in}B_{1}\cap\Omega,|D^{2}u|\le K & \text{a.e. in}B_{1}\backslash\Omega,
\end{cases} \end{equation*} where $\Omega$ is an unknown open set and $K>0$. In
particular, structural conditions on $F$ are presented which ensure that
$W^{2,n}(B_1)$ solutions achieve the optimal $C^{1,1}(B_{1/2})$ regularity when
$f$ is H\"older continuous. Moreover, if $f$ is positive on $\overline B_1$,
Lipschitz continuous, and $\{u\neq 0\} \subset \Omega$, then we obtain local
$C^1$ regularity of the free boundary under a uniform thickness assumption on
$\{u=0\}$. Lastly, we extend these results to the parabolic setting.

26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn