BMO estimates for Hodge-Maxwell systems with discontinuous anisotropic coefficients

TL;DR

This paper establishes boundary BMO estimates for Maxwell-Hodge systems with discontinuous, anisotropic coefficients, extending regularity results to more general, less smooth coefficient classes in electromagnetic theory.

Contribution

It introduces BMO estimates for Maxwell-Hodge systems with coefficients that are only bounded measurable and in a small BMO multiplier class, which includes discontinuous anisotropic tensors.

Findings

BMO estimates hold up to the boundary for these systems.

The results apply to time-harmonic Maxwell equations with discontinuous anisotropic tensors.

The regularity assumptions on coefficients are essentially sharp.

Abstract

We prove up to the boundary $\mathrm{BMO}$ estimates for linear Maxwell-Hodge type systems for $\mathbb{R}^{N}$-valued differential $k$-forms $u$ in $n$ dimensions \begin{align*} \left\lbrace \begin{aligned} d^\ast \left( A(x) du \right) &= f &&\text{ in } \Omega, d^\ast \left( B(x) u\right) &= g &&\text{ in } \Omega, \end{aligned} \right. \end{align*} with $ \nu\wedge u$ prescribed on $\partial\Omega,$ where the coefficient tensors $A,B$ are only required to be bounded measurable and in a class of `small multipliers of BMO'. This class neither contains nor is contained in $C^{0}.$ Since the coefficients are allowed to be discontinuous, the usual Korn's freezing trick can not be applied. As an application, we show BMO estimates hold for the time-harmonic Maxwell system in dimension three for a class of discontinuous anisotropic permeability and permittivity tensors. The regularity assumption on the coefficient is essentially sharp.