On traces for functional spaces related to Maxwell's equations Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications

Hodge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L 2 tangential fields and then the attention is focused on some particular Sobolev spaces of order $‐{1\over 2}$\nopagenumbers\end . In order to reach this goal, it is required to properly define the first order differential operators and to investigate their properties. When the manifold Γ is the boundary of a polyhedron Ω, these spaces are important in the analysis of tangential trace mappings for vector fields in H ( curl , Ω) on the whole boundary or on a part of it. By means of these Hodge decompositions, one can then provide a complete characterization of these trace mappings: general extension theorems, from the boundary, or from a part of it, to the inside; definition of suitable dualities and validity of integration by parts formulae. Copyright © 2001 John Wiley & Sons, Ltd.