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

$$ \mathscr{F}(w, \Omega)=\int_{\Omega}[F(x, D w(x))-f(x) \cdot w(x)] d x $$

with non-autonomous integrand $F(x, \xi)$ 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 $x \mapsto D_{\xi} F(x, \xi)$ has weak derivatives in some Lebesgue space $L^q$ and the datum $f$ is assumed to belong to a suitable Lebesgue space $L^r$. We also prove that it is possible to weaken the assumption on the datum $f$ and on the map $x \mapsto D_{\xi} F(x, \xi)$, if the minimizers are assumed to be a priori bounded.