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General solution of the inhomogenous div-curl system and consequences

Briceyda B. Delgado, R. Michael Porter

Abstract

We consider the inhomogeneous div-curl system (i.e.\ to find a vector field with prescribed div and curl) in a bounded star-shaped domain in $\mathbb{R}^3$. An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation $DW=(Df/f)\overline W$ in $\mathbb{R}^3$, such as finding the vector part when the scalar part is known. In addition, using the general solution to the div-curl system and the known existence of the solution of the inhomogeneous conductivity equation, we prove the existence of solutions of the inhomogeneous double curl equation, and give an explicit solution for the case of static Maxwell's equations with only variable permeability.

General solution of the inhomogenous div-curl system and consequences

Abstract

We consider the inhomogeneous div-curl system (i.e.\ to find a vector field with prescribed div and curl) in a bounded star-shaped domain in . An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation in , such as finding the vector part when the scalar part is known. In addition, using the general solution to the div-curl system and the known existence of the solution of the inhomogeneous conductivity equation, we prove the existence of solutions of the inhomogeneous double curl equation, and give an explicit solution for the case of static Maxwell's equations with only variable permeability.

Paper Structure

This paper contains 15 sections, 29 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Quaternionic analysis focused on $\mathbb{R}^3$
  3. Monogenic functions
  4. Standard integral operators
  5. Components of the Teodorescu operator
  6. Solution to the div-curl system
  7. Div-curl system with boundary data
  8. Application to the main Vekua equation and conductivity equation on $\mathbb{R}^3$
  9. The main Vekua equation and equivalent formulations
  10. Completion of Vekua solutions from partial data
  11. A result on Vekua-type operators
  12. Vekua boundary value problems
  13. Equation of double curl type
  14. Generalized solutions of the Maxwell system
  15. Variational methods for double curl boundary value problems

Key Result

Proposition 2.1

Let $\vec{w}\in\hbox{Har}\,(\Omega,\mathbb{R}^3)$ where $\Omega$ is simply connected. A necessary and sufficient condition for there to exist $w\in\mathfrak{M}(\Omega)$ such that $\hbox{Vec}\, w=\vec{w}$ is that $\hbox{div}\,\vec{w}=0$.

Theorems & Definitions (51)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4: GuSpr1990
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 41 more