Mixed boundary-value problems for Maxwell’s equations

Abstract

We study the Maxwell system with mixed boundary conditions in a Lipschitz domain Ω in ℝ 3 . It is assumed that two disjoint, relatively open subsets ∑ e , ∑ h of ∂Ω such that ∑ e ∩ ∑ h = ∂∑ e = ∂∑ h have been fixed, and one prescribes the tangential components of the electric and magnetic fields on ∑ e and ∑ h , respectively. Under suitable geometric assumptions on ∂Ω, ∑ e and ∑ 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.