Properties of differential operators with vanishing coefficients

In this paper, we investigate the properties of linear operators defined on $L^p(\Omega)$ that are the composition of differential operators with functions that vanish on the boundary $\partial \Omega$. We focus on bounded domains $\Omega \subset \mathbb{R}^d$ with Lipshitz continuous boundary. In this setting we are able to characterize the spectral and Fredholm properties of a large class of such operators. This includes operators of the form $Lu = \text{div}( \Phi \nabla u)$ where $\Phi$ is a matrix valued function that vanishes on the boundary, as well as operators of the form $Lu = D^{\alpha} (\varphi u)$ or $L = \varphi D^{\alpha} u$ for some function $\varphi \in \mathscr{C}^1(\bar{\Omega})$ that vanishes on $\partial \Omega$.