Open AccessOn $L^1$-Matrices with Degenerate Spectrum and Weak Convergence in Associated Weighted Sobolev Spaces
Author(s) -
Peter I. Kogut,
T.N. Rudyanova
Publication year - 2012
Publication title -
researches in mathematics
Language(s) - English
Resource type - Journals
eISSN - 2664-5009
pISSN - 2664-4991
DOI - 10.15421/241219
Subject(s) - omega , sobolev space , mathematics , standard probability space , eigenvalues and eigenvectors , degenerate energy levels , lebesgue measure , combinatorics , spectrum (functional analysis) , lebesgue integration , compact space , measure (data warehouse) , lp space , pure mathematics , physics , banach space , quantum mechanics , database , computer science
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.