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

We propose some general growth conditions on the function f = f (x, ξ), 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 \intΩ f (x, Du) dx is locally Lipschitz continuous in Ω. 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 (x, ξ) as modulus of ξ goes to +∞; 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 non-uniform elliptic variational problems to a context of uniform ellipticity.