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

We prove that local minimizers of the functionalF(u):=∫Ω|Du|p(x)log(e+|Du|)dx are of class C0,α for every 0<α<1, if the exponent p(x)>1 has logarithmic modulus of continuity. Moreover, in case the exponent p(x)>1 is a Hölder continuous function, we establish that minimizers of F(u) are of class C1,α, for some 0<α<1.