BMO estimates for Hodge–Maxwell systems with discontinuous anisotropic coefficients

We prove up to the boundary $\mathrm{BMO}$ estimates for linear Maxwell–Hodge type systems for $R^N$-valued differential $k$-forms $u$ in $n$ dimensions $\left(\begin{array}{cccc}{d}^{\ast }\left(A\left(x\right)du\right)& =f& & \text{in}\Omega ,\\ d^{\ast }\left(B\left(x\right)u\right)& =g& & \text{in}\Omega ,\end{array}\right)$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 cannot 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.