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

Abstract

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 [Formula: see text] which is convex with respect to the gradient variable, under [Formula: see text]-growth conditions, with [Formula: see text]. The main novelty here is that the results are obtained assuming that the partial map [Formula: see text] has weak derivatives in some Lebesgue space [Formula: see text] and the datum [Formula: see text] is assumed to belong to a suitable Lebesgue space [Formula: see text]. We also prove that it is possible to weaken the assumption on the datum [Formula: see text] and on the map [Formula: see text], if the minimizers are assumed to be a priori bounded.

Keywords

• Convex functionals
• a priori bounded minimizers
• higher differentiability