Lipschitz bounds and nonautonomous integrals

Abstract: We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and yield new, optimal regularity criteria even in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.