Higher differentiability results for solutions to a class of non-homogeneous elliptic problems under sub-quadratic growth conditions.

We prove a sharp higher differentiability result for local minimizers of functionals of the form F(w, Ω) = ∫_Ω [F(x,Dw(x)) - f(x) · w(x)]dx with non-autonomous integrand F(x, ξ) which is convex with respect to the gradient variable, under p-growth conditions, with 1 < p < 2. The main novelty here is that the results are obtained assuming that the partial map x ... D_ξF(x, ξ) has weak derivatives in some Lebesgue space Lq and the datum f is assumed to belong to a suitable Lebesgue space Lr. We also prove that it is possible to weaken the assumption on the datum f and on the map x ... D_ξF(x, ξ), if the minimizers are assumed to be a priori bounded. [ABSTRACT FROM AUTHOR]