On $L^1$-Matrices with Degenerate Spectrum and Weak Convergence in Associated Weighted Sobolev Spaces

We study the compactness property of the weak convergence in variable Sobolev spaces of the following sequences $\left\{ (A_n,u_n) \in L^1(\Omega; {\mathbb{R}}^{N\times N}) \times W_{A_n}(\Omega; {\Gamma}_D) \right\}$, where the squared symmetric matrices $A\colon \Omega \rightarrow {\mathbb{R}}^{N\times N}$ belong to the Lebesgue space $L^1(\Omega; {\mathbb{R}}^{N\times N})$ and their eigenvalues may vanish on subdomains of $\Omega$ with zero Lebesgue measure.