Regularity results for a priori bounded minimizers of non-autonomous functionals with discontinuous coefficients

We prove the higher differentiability and the higher integrability of the a priori bounded local minimizers of integral functionals of the form ℱ ⁢ ( v , Ω ) = ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) ⁢ d x , F(v,Ω)=∫Ωf(x,Dv(x))\,{d}x, with convex integrand satisfying p-growth conditions with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x-variable belongs to a suitable Sobolev space. The a priori boundedness of the minimizers allows us to obtain the higher differentiability under a Sobolev assumption which is independent on the dimension n and that, in the case p ≤ n - 2 {p≤ n-2} , improves previous known results. We also deal with solutions of elliptic systems with discontinuous coefficients under the so-called Uhlenbeck structure. In this case, it is well known that the solutions are locally bounded and therefore we obtain analogous regularity results without the a priori boundedness assumption.