A Remark on the Regularity of Solutions of Maxwell's Equations on Lipschitz Domains

Let ~ u be a vector eld on a bounded Lipschitz domain in R 3 , and let ~ u together with its divergence and curl be square integrable. If either the normal or the tangential component of ~ u is square inte-grable over the boundary, then ~ u belongs to the Sobolev space H 1=2 on the domain. This result gives a simple explanation for known results on the compact embedding of the space of solutions of Maxwell's equations on Lipschitz domains into L 2. Let R 3 be a bounded simply connected domain with connected Lip-schitz boundary ?. This means that ? can be represented locally as the graph of a Lipschitz function. For properties of Lipschitz domains, see 7], 3], 2]. In particular, ? has the strict cone property. We consider real vector elds ~ u on satisfying in the distributional sense We denote the inner product in L 2 (() by (;).