  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 general $(p,q)$-growth. Optimal regularity
theory for the above non-autonomous problems is a long-standing issue; the
classical approach by Giaquinta and Giusti involves assuming that the
nonlinearity $F$ 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 $F$ 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 the ratio
$\frac qp$ of 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.

Analysis of PDEs