Higher differentiability for solutions to stationary $p$-Stokes systems under sub-quadratic growth conditions

Abstract

We consider weak solutions $(u,\pi):\mathbb{R}^n\supset\Omega\to\ \mathbb{R}^n\times\ \mathbb{R}$ to stationary $p$-Stokes systems of the type \[ \begin{cases} -\mathrm{div} (a(\mathcal{E} u))+\nabla\pi=f \\ \mathrm{div}(u)=0, \end{cases} \] in $\Omega,$ where the function $a(\xi)$ satisfies $p$-growth conditions in $\xi$. By $\mathcal{E} u$ we denote the symmetric part of the gradient $Du$. In this setting, we establish results on the fractional higher differentiability of both the symmetric part of the gradient $ D u$ and of the pressure $\pi$.