In this paper we prove an estimate for the measure of superlevel sets of weak solutions to quasilinear elliptic systems in divergence form. In some special cases, such an estimate allows us to improve on the integrability of the solution.

The paper deals with very weak solutions $u\in \theta +W_0^{1,r}\left(\Omega \right)$, $max\{1,p-1\}<r<p<n$, to boundary value problems of the $p$-harmonic equation $$\left\{\begin{array}{cc}-\text{div}\left(\right|\nabla u\left(x\right){|}^{p-2}\nabla u\left(x\right))=0,\hfill & x\in \Omega ,\hfill \\ u\left(x\right)=\theta \left(x\right),\hfill & x\in \partial \Omega .\hfill \end{array}\right.(*)$$ We show that, under the assumption $\theta \in W^{1,q}\left(\Omega \right)$, $q>r$, any very weak solution $u$ to the boundary value problem ($*$) is integrable with $$u\in \left\{\begin{array}{cc}\theta +L_{\mathrm{weak}}^{q^{*}}\left(\Omega \right)\hfill & \text{for}q<n,\hfill \\ \theta +L_{\mathrm{weak}}^{\tau }\left(\Omega \right)\hfill & \text{for}q=n\text{and}\text{any}\tau <\infty ,\hfill \\ \theta +L^{\infty }\left(\Omega \right)\hfill & \text{for}q>n,\hfill \end{array}\right.$$ provided that $r$ is sufficiently close to $p$.

In this paper we consider weak solutions $𝐮:\Omega \to {ℝ}^d$ to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain $\Omega \subset {ℝ}^d$ ($d=2$ or $d=3$). For the critical case $q=\frac{3d}{d+2}$ we prove the higher integrability of $\nabla 𝐮$ which forms the basis for applying the method of differences in order to get fractional differentiability of $\nabla 𝐮$. From this we show the existence of second order weak derivatives of $u$.