Weighted estimates for Hodge-Maxwell systems

Abstract

We establish up to the boundary regularity estimates in weighted $L^{p}$ spaces with Muckenhoupt weights $A_{p}$ for weak solutions to the Hodge systems \begin{align*} d^{\ast}\left(Ad\omega\right) + B^{\intercal}dd^{\ast}\left(B\omega\right) = \lambda B\omega + f \quad \text{ in } \Omega \end{align*} with either $\nu \wedge \omega $ and $\nu \wedge d^{\ast}\left(B\omega\right)$ or $\nu \lrcorner B\omega$ and $\nu \lrcorner Ad\omega$ prescribed on $\partial\Omega.$ As a consequence, we prove the solvability of Hodge-Maxwell systems and derive Hodge decomposition theorems in weighted Lebesgue spaces. Our proof avoids potential theory, does not rely on representation formulas and instead uses decay estimates in the spirit of `Campanato method' to establish weighted $L^{p}$ estimates.