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 $u$.

$\Gamma$-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness $\epsilon$ approaches zero of a ferromagnetic thin structure ${\Omega }_{\epsilon }=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {ℝ}^2$, whose energy is given by $${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\mathrm{d}x$$ subject to $$\text{div}(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }})=0\text{on}{ℝ}^3,$$ and to the constraint $$|\overline{m}|=1\text{on}{\Omega }_{\epsilon },$$ where $W$ is any continuous function satisfying $p$-growth assumptions with $p\>1$. Partial results are also obtained in the case $p=1$, under an additional...

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 ${\int }_0^1{x}^{\alpha }{(1-x)}^{\beta }{Q}^{\gamma }\left(x\right)\mathrm{d}x=1$ $(\alpha ,\beta ,\gamma \in ℝ,\gamma \ne 0)$.