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arXiv:2403.18387v1 Announce Type: new
Abstract: 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 d\omega \right) + B^{\intercal}d d^{\ast} \left( B\omega \right) = \lambda B\omega + f \text{ in } \Omega,
\end{equation*}
with either $ \nu\wedge \omega$ and $\nu\wedge d^{\ast} \left( B\omega \right)$ or $\nu\lrcorner B\omega$ and
$\nu\lrcorner \left( A d\omega \right)$ prescribed on $\partial\Omega.$ 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.
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