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arXiv · 1504.03157

Properties of differential operators with vanishing coefficients

Daniel Jordon
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Abstract

In this paper, we investigate the properties of linear operators defined on $L^p(Ω)$ that are the composition of differential operators with functions that vanish on the boundary $\partial Ω$. We focus on bounded domains $Ω\subset \mathbb{R}^d$ 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 $Lu = \text{div}( Φ\nabla u)$ where $Φ$ is a matrix valued function that vanishes on the boundary, as well as operators of the form $Lu = D^α (φu)$ or $L = φD^α u$ for some function $φ\in \mathscr{C}^1(\barΩ)$ that vanishes on $\partial Ω$.

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Daniel Jordon. 2015-04-13. Properties of differential operators with vanishing coefficients. https://arxiv.org/abs/1504.03157

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