Holder continuity of local minimizers of vectorial integral functionals

We study the regularity of vector-valued local minimizers in W 1,p, p > 1, of the integral functional u → ∫ Ω [(μ2 + |Du|2)p/2 + f(x, u, |Du|)] dx, where Ω is an open set in ℝN and f is a continuous function, convex with respect to the last variable, such that 0 ≤ f(x, u, t) ≤ C(1 + tp). We prove that if f = f(x, t), or f = f(x, u, t) and p ≥ N, then local minimizers are locally Hölder continuous for any exponent less than 1. If f = f(x, u, t) and p < N then local minimizers are Hölder continuous for every exponent less than 1 in an open set Ω0 such that the Hausdorff dimension of Ω/Ω0 is less than N - p.