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

Abstract

We propose some general growth conditions on the function , 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 $\left|\xi \right|\to +\infty$; 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.