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

Abstract

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.