Fractional Vector Calculus and the Fractional Maxwell's Equations

TL;DR

This paper develops a fractional version of Maxwell's equations using novel fractional operators and Sobolev spaces, establishing well-posedness and setting the stage for future inverse scattering problems.

Contribution

It introduces a fractional curl operator, a projection map, and fractional Sobolev spaces, enabling reformulation and analysis of fractional Maxwell's equations.

Findings

Proved well-posedness of fractional Maxwell's equations in weighted Sobolev spaces.

Established a bijection between two-point and one-point fractional fields.

Reformulated the system entirely in terms of one-point fields.

Abstract

We consider a fractional variant of Maxwell's equations, where the electric and magnetic fields are modeled as two-point fields. To formulate the system, we introduce a fractional curl operator that is compatible with the fractional divergence operator, ensuring the divergence-free condition. A key ingredient is a projection map $\Pi$ that reduces two-point fields to one-point fields. We also define a new fractional Sobolev space whose elements enjoy a fractional Helmholtz decomposition and observe that the projection $\Pi$ is a bijection in this space, which allows us to reformulate the problem entirely in terms of one-point fields. We then prove the well-posedness of the equations in one-point fields in weighted fractional Sobolev spaces, and deduce a corresponding well-posedness result for the two-points fractional Maxwell system. This constitutes a first necessary step towards the resolution of a scattering inverse problem for the fractional Maxwell's equations, which will be the topic of future work.