Morrey-Lorentz estimates for Hodge-type systems
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908292218880000 |
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| author | Sengupta, Banhirup Sil, Swarnendu |
| author_facet | Sengupta, Banhirup Sil, Swarnendu |
| contents | We prove up to the boundary regularity estimates in Morrey-Lorentz spaces for weak solutions of the linear system of differential forms with regular anisotropic coefficients
\begin{equation*}
d^{\ast} \left( A dω\right) + B^{\intercal}d d^{\ast} \left( Bω\right) = λBω+ f \text{ in } Ω,
\end{equation*}
with either $ ν\wedge ω$ and $ν\wedge d^{\ast} \left( Bω\right)$ or $ν\lrcorner Bω$ and
$ν\lrcorner \left( A dω\right)$ prescribed on $\partialΩ.$ We derive these estimates from the $L^{p}$ estimates obtained in \cite{Sil_linearregularity} in the spirit of Campanato's method. Unlike Lorentz spaces, Morrey spaces are neither interpolation spaces nor rearrangement invariant. So Morrey estimates can not be obtained directly from the $L^{p}$ estimates using interpolation. We instead adapt an idea of Lieberman \cite{Lieberman_morrey_from_Lp} to our setting to derive the estimates. Applications to Hodge decomposition in Morrey-Lorentz spaces, Gaffney type inequalities and estimates for related systems such as Hodge-Maxwell systems and `div-curl' systems are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Morrey-Lorentz estimates for Hodge-type systems Sengupta, Banhirup Sil, Swarnendu Analysis of PDEs 35J57, 35B65, 35Q60 We prove up to the boundary regularity estimates in Morrey-Lorentz spaces for weak solutions of the linear system of differential forms with regular anisotropic coefficients \begin{equation*} d^{\ast} \left( A dω\right) + B^{\intercal}d d^{\ast} \left( Bω\right) = λBω+ f \text{ in } Ω, \end{equation*} with either $ ν\wedge ω$ and $ν\wedge d^{\ast} \left( Bω\right)$ or $ν\lrcorner Bω$ and $ν\lrcorner \left( A dω\right)$ prescribed on $\partialΩ.$ We derive these estimates from the $L^{p}$ estimates obtained in \cite{Sil_linearregularity} in the spirit of Campanato's method. Unlike Lorentz spaces, Morrey spaces are neither interpolation spaces nor rearrangement invariant. So Morrey estimates can not be obtained directly from the $L^{p}$ estimates using interpolation. We instead adapt an idea of Lieberman \cite{Lieberman_morrey_from_Lp} to our setting to derive the estimates. Applications to Hodge decomposition in Morrey-Lorentz spaces, Gaffney type inequalities and estimates for related systems such as Hodge-Maxwell systems and `div-curl' systems are discussed. |
| title | Morrey-Lorentz estimates for Hodge-type systems |
| topic | Analysis of PDEs 35J57, 35B65, 35Q60 |
| url | https://arxiv.org/abs/2403.18387 |