Mixed boundary-value problems for Maxwell’s equations

Abstract:

We study the Maxwell system with mixed boundary conditions in a Lipschitz domain $\Omega$ in $\mathbb {R}^3$. It is assumed that two disjoint, relatively open subsets $\Sigma ^e$, $\Sigma ^h$ of $\partial \Omega$ such that $\overline {\Sigma ^e}\cap \overline {\Sigma ^h}=\partial \Sigma ^e=\partial \Sigma ^h$ have been fixed, and one prescribes the tangential components of the electric and magnetic fields on $\Sigma ^e$ and $\Sigma ^h$, respectively. Under suitable geometric assumptions on $\partial \Omega$, $\Sigma ^e$ and $\Sigma ^h$, we prove that this boundary value problem is well-posed when $L^p$-estimates for the nontangential maximal function are sought, with $p$ near $2$. A higher-dimensional version of this result is established as well, in the language of differential forms. This extends earlier work by R. Brown and by the author and collaborators.