Fractional Vector Calculus and the Fractional Maxwell's Equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917462516170752 |
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| author | Covi, Giovanni Wu, Ruirui |
| author_facet | Covi, Giovanni Wu, Ruirui |
| contents | 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 $Π$ 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 $Π$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04241 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractional Vector Calculus and the Fractional Maxwell's Equations Covi, Giovanni Wu, Ruirui Analysis of PDEs 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 $Π$ 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 $Π$ 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. |
| title | Fractional Vector Calculus and the Fractional Maxwell's Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.04241 |