A general regularity theorem for functionals with φ-growth

We prove C 1 , α -regularity for local minimizers of functionals ∫ Ω φ ( | ∇ u | ) d x , where φ is a Young function. In order to generalize the results of Fuchs (2011) [9] and Diening et al. (2009) [6], we assume for the function φ only the ( Δ 2 ) -condition and φ ″ ( t ) ⩾ ϵ ˆ φ ′ ( t ) t for a positive constant ϵ ˆ (and of course a Hölder-condition for the second derivatives).