Singularities of Electromagnetic Fields¶in Polyhedral Domains

Abstract:

In this paper, we investigate the singular solutions of time-harmonic Maxwell equations in a domain which has edges and polyhedral corners. It is now well known that in the presence of non-convex edges, the solution fields have no square integrable gradients in general and that the main singularities are the gradients of singular functions of the Laplace operator [4,–5]. We show how this type of result can be derived from the classical Mellin analysis, and how this analysis leads to sharper results concerning the singular parts which belong to H ¹. For the singular functions, we exhibit simple and explicit formulas based on (generalized) Dirichlet and Neumann singularities for the Laplace operator. These formulas are more explicit than the results announced in our note [10].