Morrey-Lorentz estimates for Hodge-type systems
Banhirup Sengupta, Swarnendu Sil
TL;DR
We address boundary regularity for the linear Hodge-type system $d^{\ast}(A d\omega) + B^{\intercal} d d^{\ast}(B\omega) = \lambda B\omega + f$ in a bounded domain, proving up-to-boundary Morrey-Lorentz estimates for weak solutions with anisotropic coefficients. The authors adapt Campanato-style techniques to Morrey-Lorentz spaces via Lorentz decay estimates, following and modifying Lieberman’s program to overcome the non-interpolation nature of Morrey spaces. The results include a comprehensive Hodge decomposition in Morrey-Lorentz spaces, and parallel estimates for Maxwell, div-curl, and Gaffney-type inequalities, all with explicit boundary data handled. This framework extends the regularity theory to Morrey-Lorentz scales, enabling precise boundary control and decomposition in contexts with anisotropic coefficients and broader function spaces.
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.