  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.

Analysis of PDEs