Regularity results for a class of functionals with non-standard growth

We consider the integral functional integral f(x, Du) dx under non-standard growth assumptions that we call p(x) type: namely, we assume that \z\(p(x)) less than or equal to f (x, z) less than or equal to L (1 + \z\(p(x))), a relevant model case being the functional integral \Du\(p(x)) dx. Under sharp assumptions on the continuous function p(x) > 1 we prove regularity of minimizers. Energies exhibiting this growth appear in several models from mathematical physics.