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

## TL;DR

This paper derives a general solution for a complex curl equation with a parameter in three-dimensional domains using quaternionic analysis, and applies it to boundary value problems and Maxwell's equations in chiral media.

## Contribution

It introduces a new right inverse operator for the curl+ operator and applies it to solve boundary value problems and Maxwell equations in complex media.

## Key findings

- Explicit solution for curl+ equation in bounded domains.
- Application to Neumann boundary value problems.
- Application to time-harmonic Maxwell system in chiral media.

## 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.