Spectral properties of the gradient operator with nonconstant coefficients

TL;DR

This paper studies the spectral properties of a gradient operator with nonconstant positive coefficients in Lipschitz domains, focusing on the $S$-spectrum and resolvent estimates within Clifford algebra frameworks.

Contribution

It identifies spectral regions and estimates for the gradient operator with variable coefficients, extending spectral theory to Clifford modules and nonconstant coefficient operators.

Findings

Identifies bisectorial and strip-type regions in the $S$-resolvent set.

Provides estimates for the resolvent operator of the gradient operator.

Links spectral properties to boundary conditions via the operator $Q_s(T)$.

Abstract

In mathematical physics, the gradient operator with nonconstant coefficients encompasses various models, including Fourier's law for heat propagation and Fick's first law, that relates the diffusive flux to the gradient of the concentration. Specifically, consider $n\geq 3$ orthogonal unit vectors $e_1,\dots,e_n\in\mathbb{R}^n$, and let $\Omega\subseteq\mathbb{R}^n$ be some (in general unbounded) Lipschitz domain. This paper investigates the spectral properties of the gradient operator $T=\sum_{i=1}^ne_ia_i(x)\frac{\partial}{\partial x_i}$ with nonconstant positive coefficients $a_i:\overline{\Omega}\to(0,\infty)$. Under certain regularity and growth conditions on the $a_i$, we identify bisectorial or strip-type regions that belong to the $S$-resolvent set of $T$. Moreover, we obtain suitable estimates of the associated resolvent operator. Our focus lies in the spectral theory on the $S$-spectrum, designed to study the operators acting in Clifford modules $V$ over the Clifford algebra $\mathbb{R}_n$, with vector operators being a specific crucial subclass. The spectral properties related to the $S$-spectrum of $T$ are linked to the inversion of the operator $Q_s(T):=T^2-2s_0T+|s|^2$, where $s\in\mathbb{R}^{n+1}$ is a paravector, i.e., it is of the form $s=s_0+s_1e_1+\dots+s_ne_n$. This spectral problem is substantially different from the complex one, since it allows to associate general boundary conditions to $Q_s(T)$, i.e., to the squared operator $T^2$.