BMO estimates for Hodge-Maxwell systems with discontinuous anisotropic coefficients
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914898314788864 |
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| author | Kumar, Dharmendra Sil, Swarnendu |
| author_facet | Kumar, Dharmendra Sil, Swarnendu |
| contents | We prove up to the boundary $\mathrm{BMO}$ estimates for linear Maxwell-Hodge type systems for $\mathbb{R}^{N}$-valued differential $k$-forms $u$ in $n$ dimensions
\begin{align*}
\left\lbrace \begin{aligned}
d^\ast \left( A(x) du \right) &= f &&\text{ in } Ω,
d^\ast \left( B(x) u\right) &= g &&\text{ in } Ω,
\end{aligned}
\right.
\end{align*}
with $ ν\wedge u$ prescribed on $\partialΩ,$ where the coefficient tensors $A,B$ are only required to be bounded measurable and in a class of `small multipliers of BMO'. This class neither contains nor is contained in $C^{0}.$ Since the coefficients are allowed to be discontinuous, the usual Korn's freezing trick can not be applied. As an application, we show BMO estimates hold for the time-harmonic Maxwell system in dimension three for a class of discontinuous anisotropic permeability and permittivity tensors. The regularity assumption on the coefficient is essentially sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_06615 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | BMO estimates for Hodge-Maxwell systems with discontinuous anisotropic coefficients Kumar, Dharmendra Sil, Swarnendu Analysis of PDEs 35J57 We prove up to the boundary $\mathrm{BMO}$ estimates for linear Maxwell-Hodge type systems for $\mathbb{R}^{N}$-valued differential $k$-forms $u$ in $n$ dimensions \begin{align*} \left\lbrace \begin{aligned} d^\ast \left( A(x) du \right) &= f &&\text{ in } Ω, d^\ast \left( B(x) u\right) &= g &&\text{ in } Ω, \end{aligned} \right. \end{align*} with $ ν\wedge u$ prescribed on $\partialΩ,$ where the coefficient tensors $A,B$ are only required to be bounded measurable and in a class of `small multipliers of BMO'. This class neither contains nor is contained in $C^{0}.$ Since the coefficients are allowed to be discontinuous, the usual Korn's freezing trick can not be applied. As an application, we show BMO estimates hold for the time-harmonic Maxwell system in dimension three for a class of discontinuous anisotropic permeability and permittivity tensors. The regularity assumption on the coefficient is essentially sharp. |
| title | BMO estimates for Hodge-Maxwell systems with discontinuous anisotropic coefficients |
| topic | Analysis of PDEs 35J57 |
| url | https://arxiv.org/abs/2310.06615 |