On the curl operator and some characterizations of matrix fields in Lipschitz domains

Abstract

As well-known, De Rham's Theorem is a classical way to characterize vector fields as the gradient of the scalar fields, it is a tool of great importance in the theory of fluids mechanic. The first aim of this paper is to provide a useful rotational version of this theorem to establish several results on boundary value problems in the field of electromagnetism. Gurtin presented the completeness of Beltrami's representation in smooth domains for smooth symmetric matrix. The extension of Beltrami's completeness was obtained by Geymonat and Krasucki when the domain Ω is only Lipschitz and for symmetric matrix in Ls2(Ω). Our second aim is to present new extensions of Beltrami's completeness results on Lipschitz domains, firstly in the case where the data are in Ds(Ω) and secondly when the data are in Wsm,r(Ω). The third objective is to extend Saint-Venant's Theorem to the case of distributions.