Let ${\lambda }_1\left(Q\right)$ be the first eigenvalue of the Sturm-Liouville problem $$y^{\text{'}\text{'}}-Q\left(x\right)y+\lambda y=0,y\left(0\right)=y\left(1\right)=0,0<x<1.$$ We give some estimates for $m_{\alpha ,\beta ,\gamma }={inf}_{Q\in T_{\alpha ,\beta ,\gamma }}{\lambda }_1\left(Q\right)$ and $M_{\alpha ,\beta ,\gamma }={sup}_{Q\in T_{\alpha ,\beta ,\gamma }}{\lambda }_1\left(Q\right)$, where $T_{\alpha ,\beta ,\gamma }$ is the set of real-valued measurable on $\left[0,1\right]$ $x^{\alpha }{(1-x)}^{\beta }$-weighted $L_{\gamma }$-functions $Q$ with non-negative values such that ${\int }_0^1{x}^{\alpha }{(1-x)}^{\beta }{Q}^{\gamma }\left(x\right)\mathrm{d}x=1$ $(\alpha ,\beta ,\gamma \in ℝ,\gamma \ne 0)$.

As a measure of deformation we can take the difference $D\overrightarrow{\phi }-R$, where $D\overrightarrow{\phi }$ is the deformation gradient of the mapping $\overrightarrow{\phi }$ and $R$ is the deformation gradient of the mapping $\overrightarrow{\gamma }$, which represents some proper rigid motion. In this article, the norm $\parallel D\overrightarrow{\phi }{-R\parallel }_{L^p\left(\Omega \right)}$ is estimated by means of the scalar measure $e\left(\overrightarrow{\phi }\right)$ of nonlinear strain. First, the estimates are given for a deformation $\overrightarrow{\phi }\in W^{1,p}\left(\Omega \right)$ satisfying the condition $\overrightarrow{\phi }{|}_{\partial \Omega }=\overrightarrow{\mathrm{id}}$. Then we deduce the estimate in the case that $\overrightarrow{\phi }\left(x\right)$ is a bi-Lipschitzian deformation and $\overrightarrow{\phi }{|}_{\partial \Omega }\ne \overrightarrow{\mathrm{id}}$.

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 $u$.