Higher integrability of the Lorentz force for weak solutions to Maxwell's equations in complex geometries

Abstract

We consider the stationary Maxwell system in a domain filled with different materials. The magnetic permeability being only piecewise smooth, we have to take into account the natural interface conditions for the electromagnetic fields. We present two sets of hypotheses under which we can prove the existence of weak solutions to the Maxwell system such that the Lorentz force jxB is integrable to a power larger than 6/5. This property is important for the investigation of problems in magnetohydrodynamics, with many industrial applications such as crystal growth.