Equations of Maxwell Type

TL;DR

This paper develops an abstract framework for Maxwell-like equations derived from elliptic complexes, analyzing boundary value problems and the Cauchy problem with a formal theory akin to classical Maxwell equations.

Contribution

It introduces a new system of equations based on elliptic complexes, extending Maxwell's equations abstractly and studying their boundary value problems.

Findings

Formal similarity to classical Maxwell equations

Analysis of boundary value problems for the abstract system

Focus on the Cauchy problem in the abstract setting

Abstract

For an elliptic complex of first order differential operators on a smooth manifold, we define a system of two equations which can be thought of as abstract Maxwell equations. The formal theory of this system proves to be very similar to that of classical Maxwell's equations. The paper focuses on boundary value problems for the abstract Maxwell equations, especially on the Cauchy problem.