BMO estimates for Hodge-Maxwell systems with discontinuous anisotropic coefficients

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.