Weighted estimates for Hodge-Maxwell systems

TL;DR

This paper develops weighted regularity estimates for solutions to Hodge-Maxwell systems in bounded domains, proving solvability and Hodge decompositions in weighted Lebesgue spaces without relying on potential theory.

Contribution

It introduces a novel approach using decay estimates and the Campanato method to establish weighted $L^{p}$ estimates for Hodge-Maxwell systems, avoiding traditional potential theory techniques.

Findings

Established boundary regularity estimates in weighted $L^{p}$ spaces.

Proved solvability of Hodge-Maxwell systems in weighted Lebesgue spaces.

Derived Hodge decomposition theorems in weighted spaces.

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.