Unified a-priori estimates for minimizers under $p,q-$growth and exponential growth

Abstract: We propose some general growth conditions on the function $% f=f\left( x,\xi \right) $, including the so-called natural growth, or polynomial, or $p,q-$growth conditions, or even exponential growth, in order to obtain that any local minimizer of the energy integral $\;\int_{\Omega }f\left( x,Du\right) dx\,$ is locally Lipschitz continuous in $\Omega $. In fact this is the fundamental step for further regularity: the local boundedness of the gradient of any Lipschitz continuous local minimizer a-posteriori makes irrelevant the behavior of the integrand $f\left( x,\xi \right) $ as $\left\vert \xi \right\vert \rightarrow +\infty $; i.e., the general growth conditions a posteriori are reduced to a standard growth, with the possibility to apply the classical regularity theory. In other words, we reduce some classes of \textit{non-uniform} elliptic variational problems to a context of uniform ellipticity.