Compact Embedding Theorem Associated with Classical Weight Functions in Two Variables

For a classical weight function $\rho$ defined on a simply connected open subset $\mathrm{\Omega }$ of ${\mathbb{R}}^2$ (either bounded or unbounded) with piecewise $C^1$ boundary, we prove density and compact embedding of a matrix-weighted Sobolev space in the weighted Lebesgue space $L^2(\mathrm{\Omega },\rho )$. As an application, we investigate via a variational method, eigenvalue problem for a degenerate Helmholtz operator on triangle.