A Right Inverse Operator for $${\text {curl}}+\lambda $$ curl + λ and Applications

Abstract

A general solution of the equation $${\text {curl}}\,\vec {w}+\lambda \vec {w}=\vec {g},\,\lambda \in \mathbb {C},\,\lambda \ne 0$$ is obtained for an arbitrary bounded domain $$\Omega \subset \mathbb {R}^{3}$$ with a Liapunov boundary and $$\vec {g}\in W^{p,{\text {div}}}\left( \Omega \right) =\left\{ \vec {u}\in L^{p}\left( \Omega \right) :\,{\text {div}}\,\vec {u}\in L^{p}\left( \Omega \right) ,\,1<p<\infty \right\} $$ . 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 $${\text {curl}}\,\vec {w}+\lambda \vec {w}=\vec {g}$$ as well as to the nonhomogeneous time-harmonic Maxwell system for achiral and chiral media.