A right inverse operator for $\mathrm{curl}+\lambda$ and applications

Abstract

A general solution of the equation $\mathrm{curl}\vec{w}+\lambda \vec{w}=\overrightarrow{g},\lambda \in \mathbb{C},\lambda \ne 0$ is obtained for an arbitrary bounded domain $\Omega \subset {\mathbb{R}}^3$ with a Liapunov boundary and $\overrightarrow{g}\in W^{p,\mathrm{div}}\left(\Omega \right)=\left\{\overrightarrow{u}\in L^p\left(\Omega \right):\mathrm{div}\overrightarrow{u}\in L^p\left(\Omega \right),1<p<\infty \right\}$. The result is based on the use of classical integral operators of quaternionic analysis. Applications of the main result are considered to a Neumann boundary value problem for the equation $\mathrm{curl}\vec{w}+\lambda \vec{w}=\overrightarrow{g}$ as well as to the nonhomogeneous time-harmonic Maxwell system for achiral and chiral media.