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

TL;DR

This paper establishes higher differentiability of solutions to certain non-homogeneous elliptic problems with sub-quadratic growth, under minimal regularity assumptions on the data and integrand.

Contribution

It introduces new higher differentiability results for minimizers with non-autonomous integrands under weak regularity conditions, extending previous theories.

Findings

Higher differentiability results for p-growth with 1<p<2

Weak regularity assumptions on the integrand's derivatives

Results hold under boundedness assumptions on minimizers

Abstract

We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,\Omega\right)=\int_{\Omega}\left[ F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ 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.