Dispersive Estimates for Maxwell's Equations in the Exterior of a Sphere
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909324507348992 |
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| author | Waters, Alden Fang, Yan-Long |
| author_facet | Waters, Alden Fang, Yan-Long |
| contents | The goal of this article is to establish general principles for high frequency dispersive estimates for Maxwell's equation in the exterior of a perfectly conducting ball. We construct entirely new generalized eigenfunctions for the corresponding Maxwell propagator. We show that the propagator corresponding to the electric field has a global rate of decay in $L^1-L^{\infty}$ operator norm in terms of time $t$ and powers of $h$. In particular we show that some, but not all, polarizations of electromagnetic waves scatter at the same rate as the usual wave operator. The Dirichlet Laplacian wave operator $L^1-L^{\infty}$ norm estimate should not be expected to hold in general for Maxwell's equations in the exterior of a ball because of the Helmholtz decomposition theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_00536 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dispersive Estimates for Maxwell's Equations in the Exterior of a Sphere Waters, Alden Fang, Yan-Long Analysis of PDEs Functional Analysis Spectral Theory The goal of this article is to establish general principles for high frequency dispersive estimates for Maxwell's equation in the exterior of a perfectly conducting ball. We construct entirely new generalized eigenfunctions for the corresponding Maxwell propagator. We show that the propagator corresponding to the electric field has a global rate of decay in $L^1-L^{\infty}$ operator norm in terms of time $t$ and powers of $h$. In particular we show that some, but not all, polarizations of electromagnetic waves scatter at the same rate as the usual wave operator. The Dirichlet Laplacian wave operator $L^1-L^{\infty}$ norm estimate should not be expected to hold in general for Maxwell's equations in the exterior of a ball because of the Helmholtz decomposition theorem. |
| title | Dispersive Estimates for Maxwell's Equations in the Exterior of a Sphere |
| topic | Analysis of PDEs Functional Analysis Spectral Theory |
| url | https://arxiv.org/abs/2308.00536 |