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 of the space of bivariate polynomials and compact embedding of the matrix-weighted Sobolev space $W^{1,2}(\Omega,\rho,\rho\Phi)$ 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.