Compact Embedding Theorem Associated with Classical Weight {Functions} in Two Variables

TL;DR

This paper proves density and compact embedding of matrix-weighted Sobolev spaces in weighted Lebesgue spaces for classical weights in two variables, and applies it to eigenvalue problems of degenerate Helmholtz operators.

Contribution

It establishes new density and compact embedding results for matrix-weighted Sobolev spaces associated with classical weights in two variables.

Findings

Proves density of matrix-weighted Sobolev spaces in weighted Lebesgue spaces.

Establishes compact embedding results for these Sobolev spaces.

Applies results to eigenvalue problems of degenerate Helmholtz operators on triangles.

Abstract

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 of a degenerate Helmholtz operator on triangle.