Compact Embedding Theorem Associated with Classical Weight Functions in Two Variables

Abstract

For a classical weight function $\ddot{I}$ defined on a simply connected open subset $\hat{I}\text{c◯}$ 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 $W^{1,2}(\hat{I}\text{c◯},\ddot{I},\ddot{I}\hat{I}¦)$ in the weighted Lebesgue space $L^2(\hat{I}\text{c◯},\ddot{I})$. As an application, we investigate via a variational method, eigenvalue problem for a degenerate Helmholtz operator on triangle.