Properties of differential operators with vanishing coefficients

Abstract

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 $Lu=D^{\alpha }(\phi u)$ or $L=\phi D^{\alpha }u$ for some function $\phi \in {\mathcal{C}}^1(\bar{\Omega })$ that vanishes on $\partial \Omega$.