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

Abstract

A general solution of the equation $\operatorname{curl}\vec{w}+\lambda\vec {w}=\overrightarrow{g},\,\lambda\in\mathbb{C},\,\lambda\neq0$ is obtained for an arbitrary bounded domain $\Omega\subset\mathbb{R}^{3}$ with a Liapunov boundary and $\overrightarrow{g}\in W^{p,\operatorname{div}}\left( \Omega\right) =\left\{ \overrightarrow{u}\in L^{p}\left( \Omega\right) :\,\operatorname{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 $\operatorname{curl}\vec{w}+\lambda\vec {w}=\overrightarrow{g}$ as well as to the nonhomogeneous time-harmonic Maxwell system for achiral and chiral media.