Compact Embedding Theorem Associated with Classical Weight Functions in Two Variables

For a classical weight function $\rho$ defined on a simply connected open subset $\Omega$ 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 $L^2(\Omega,\, \rho)$. As an application, we investigate via a variational method, eigenvalue problem for a degenerate Helmholtz operator on triangle.