Properties of differential operators with vanishing coefficients

In this paper, we investigate the properties of linear operators defined on $L^p(\mathrm{\Omega })$ that are the composition of differential operators with functions that vanish on the boundary $\mathrm{\partial }\mathrm{\Omega }$. We focus on bounded domains $\mathrm{\Omega }\subset {\mathbb{R}}^d$ 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$L = \varphi D^{\alpha} u$forsomefunction$\varphi \in \mathscr{C}^1(\bar{\Omega})$thatvanisheson$\partial \Omega$.