  We present an extension of the classical De Giorgi class, and then we show
that functions in this new class are locally bounded and locally H\"older
continuous. Some applications are given. As a first application, we give a
regularity result for local minimizers $u:\Omega \subset \mathbb R^4
\rightarrow \mathbb R^4$ of a special class of polyconvex functionals with
splitting form in four dimensional Euclidean spaces. Under some structural
conditions on the energy density, we prove that each component $u^\alpha$ of
the local minimizer $u$ belongs to the generalized De Giorgi class, then one
can derive that it is locally bounded and locally H\"older continuous. Our
result can be applied to polyconvex integrals whose prototype is $$ \int_\Omega
\Big(\sum_{\alpha =1}^4 |Du^\alpha|^p + \sum_{\beta =1}^6 |({\rm adj}_2 Du
)^\beta | ^q +\sum_{\gamma =1}^4 |({\rm adj}_3 Du )^\gamma | ^r +|\det Du|^s
\Big ) \mathrm {d}x $$ with suitable $p,q,r,s\ge 1$. As a second application,
we consider a degenerate linear elliptic equation of the form $$ -\mbox {div}
(a(x)\nabla u)=-\mbox {div}F, $$ with $0<a(x) \le \beta <+\infty$. We prove, by
virtue of the generalized De Giorgi class, that any weak solution is locally
bounded and locally H\"older continuous provided that $\frac 1 {a(x)}$ and
$F(x)$ belong to some suitable locally integrable function spaces. As a third
application, we show that our theorem can be applied in dealing with regularity
issues of elliptic equations with non-standard grow conditions. As a fourth
application we treat with quasilinear elliptic systems. Under suitable
assumptions on the coefficients, we show that any of its weak solutions is
locally bounded and locally H\"older continuous.
