Weighted estimates for Hodge-Maxwell systems
Rohit Mahato, Swarnendu Sil
TL;DR
The paper develops a theory of up-to-boundary regularity for general Hodge systems in weighted $L^{p}$ spaces with Muckenhoupt $A_p$ weights. It introduces a Campanato-inspired decay framework together with pointwise maximal inequalities and boundary flattening to obtain weighted Hessian estimates without potential theory or representation formulas. The main contribution is a global apriori estimate in $W^{r+2,p}_{w}$ for solutions to $\,\mathfrak{L}\omega = \lambda B\omega + f$ under tangential or normal boundary data, which yields solvability and Hodge decompositions for Hodge, Hodge–Maxwell, Morrey-type, and div–curl problems in weighted spaces, along with Gaffney-type inequalities. These results extend the weighted regularity theory to the Hodge–Maxwell framework and pave the way for extrapolation to Orlicz-type spaces, with potential applications to electromagnetic theory in weighted media and related PDE 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(Adω\right) + B^{\intercal}dd^{\ast}\left(Bω\right) = λBω+ f \quad \text{ in } Ω \end{align*} with either $ν\wedge ω$ and $ν\wedge d^{\ast}\left(Bω\right)$ or $ν\lrcorner Bω$ and $ν\lrcorner Adω$ prescribed on $\partialΩ.$ 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.