Differential Forms and Boundary Integral Equations for Maxwell-Type Problems

Abstract

We present boundary-integral equations for Maxwell-type problems in a differential-form setting. Maxwell-type problems are governed by the differential equation $(\delta \mathrm{d}-k^2)\omega =0$, where $k\in \mathbb{C}$ holds, subject to some restrictions. This problem class generalizes $\text{curl}\text{curl}$- and $\mathrm{div}\text{grad}$-types of problems in three dimensions. The goal of the paper is threefold: 1) Establish the Sobolev-space framework in the full generality of differential-form calculus on a smooth manifold of arbitrary dimension and with Lipschitz boundary. 2) Introduce integral transformations and fundamental solutions, and derive a representation formula for Maxwell-type problems. 3) Leverage the power of differential-form calculus to gain insight into properties and inherent symmetries of boundary-integral equations of Maxwell-type.

69 pages

References in corpus (1)

• Differential Form Valued Forms and Distributional Electromagnetic Sources

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