Morrey-Lorentz estimates for Hodge-type systems

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 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.