Regularity under nonstandard growth conditions

Some regularity results are presented for local minimizers of integral functionals $$\int f(x,Du)dx$$ satisfying non standard growth assumptions, i.e., such that the growth exponent $q$ and the ellipticity exponent $p$ are different in the inequality $$|z{|}^p\le f(x,z)\le 1+|z{|}^q$$ Both the homogeneous-anisotropic and the inhomogeneous-isotropic cases are considered.