Regularity of weak solution to a $p$-curl-system

Abstract

In this note we study the regularity of weak solutions to a nonlinear steady-state Maxwell's equation in conductive media: $\nabla\times \big[|{\nabla \times }{{\bf H}}|^{p-2} \nabla \times {{\bf H}} \big]={ {\bf F} }(x), p>1, $ where ${{\bf H}}(x)$ represents the magnetic field while ${ {\bf F} }(x)$ is the internal magnetic current. It is shown that the weak solution to the above system is of class $C^{1+\alpha}$, which is optimal. The basic idea is to introduce a suitable potential and then to transform the system into a $p-$Laplacian equation subject to a Neumann type of boundary condition. The desired regularity is established by using the known theory for the scalar p-Laplacian equation.