Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gradient continuity estimates for the normalized $p$-Poisson equation   In this paper, we obtain gradient continuity estimates for viscosity
solutions of $\Delta_{p}^N u= f$ in terms of the scaling critical $L(n,1 )$
norm of $f$, where $\Delta_{p}^N$ is the normalized $p-$Laplacian operator
defined in (1.2) below. Our main result, Theorem 2.2, corresponds to the
borderline gradient continuity estimate in terms of the modified Riesz
potential $\tilde I^{f}_{q}$. Moreover, for $f \in L^{m}$ with $m>n$, we also
obtain $C^{1,\alpha}$ estimates, see Theorem 2.3 below. This improves one of
the regularity results in [3], where a $C^{1,\alpha}$ estimate was established
depending on the $L^{m}$ norm of $f$ under the additional restriction that
$p>2$ and $m > \text{max} (2,n, \frac{p}{2}) $ (see Theorem 1.2 in [3]). We
also mention that differently from the approach in [3], which uses methods from
divergence form theory and nonlinear potential theory in the proof of Theorem
1.2, our method is more non-variational in nature, and it is based on
separation of phases inspired by the ideas in [36]. Moreover, for $f$
continuous, our approach also gives a somewhat different proof of the $C^{1,
\alpha}$ regularity result, Theorem 1.1, in [3].

19
The categories of this paper are: eess
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol