We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla u_k\right\}$, bounded in $L^p(\Omega ;{ℝ}^{m\times n})$ if and $\Omega \subset {ℝ}^n$ is a bounded domain with the extension property in $W^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of are required and links with lower semicontinuity...

Let Ω be a smooth bounded domain in ${𝐑}^n$, n > 1, let a and f be continuous functions on $\overline{\Omega }$, $1^{☆}=\frac{n}{n-1}$. We are concerned here with the existence of solution in $BV\left(\Omega \right)$, positive or not, to the problem:
 $$\left\{\begin{array}{cccc}\hfill -\mathrm{div}\sigma +a\left(x\right)signu& amp;={f\left|u\right|}^{1^{☆}-2}u\sigma .\nabla u\hfill & amp;=\left|\nabla u\right|\mathrm{in}\Omega u\mathrm{is}\mathrm{not}\mathrm{identically}\mathrm{zero},& amp;-\sigma .n\left(u\right)=\left|u\right|\mathrm{on}\partial \Omega .\end{array}\right.$$ This problem is closely related to the extremal functions for the problem of the best constant of $W^{1,1}\left(\Omega \right)$ into $L^{\frac{N}{N-1}}\left(\Omega \right)$.