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 ∫𝛺 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 |𝜉| → +∞; 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.