Lipschitz Bounds and Nonautonomous Integrals

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 from 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 also yield new, optimal regularity criteria in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.

Automated summary

The study presents a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity, ranging from unbalanced polynomial growth conditions to fast, exponential growth. The sharp results obtained are optimal with respect to all considered data, and also provide new regularity criteria in the classical uniformly elliptic case. Furthermore, a classification of different types of nonuniform ellipticity is given, along with suitable conditions to obtain regularity theorems.