Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regularity theory for non-autonomous partial differential equations
  without Uhlenbeck structure   We establish maximal local regularity results of weak solutions or local
minimizers of \[ \operatorname{div} A(x, Du)=0 \quad\text{and}\quad \min_u
\int_\Omega F(x,Du)\,dx, \] providing new ellipticity and continuity
assumptions on $A$ or $F$ with a general $(p,q)$-growth. Optimal regularity
theory for the above nonautonomous problems is a long-standing issue; the
classical approach by Giaquinta and Giusti involves assuming that the
nonlinearity $A$ satisfies a structure condition. This means that the growth
and ellipticity conditions depend on a given special function, such as $t^p$,
$\varphi(t)$, $t^{p(x)}$, $t^p+a(x)t^q$, and not only $A$ but also the given
function is assumed to satisfy suitable continuity conditions. Hence these
regularity conditions depend on given special functions.
  In this paper we study the problem without recourse to special function
structure and without assuming Uhlenbeck structure. We introduce a new
ellipticity condition using $A$ or $F$ only, which entails that the function is
quasi-isotropic, i.e. it may depend on the direction, but only up to a
multiplicative constant. Moreover, we formulate the continuity condition on $A$
or $F$ without specific structure and without direct restriction on $\frac qp$
(the parameters from the $(p,q)$-growth condition). We establish local
$C^{1,\alpha}$-regularity for some $\alpha\in(0,1)$ and $C^{\alpha}$-regularity
for any $\alpha\in(0,1)$ of weak solutions and local minimizers. Previously
known, essentially optimal, regularity results are included as special cases.

17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc