An isomorphic property of two Hilbert spaces appearing in electromagnetism: Analysis by the mixed formulation

Abstract

Let Ω be a bounded domain inR3 with Lipschitz continuous boundary ∂Ω. In electromagnetism, we use the Hilbert spaceV(Ω) of vector-valued functions which, along with their rotations and divergences, are square summable in Ω and whose tangential components on ∂Ω vanish. In this paper, it is proven thatV(Ω) is isomorphic to {H1(Ω)}3 ∩V(Ω) when Ω is convex, whereH1(Ω) is the usual first order Sobolev space. To this end, we adopt the techniques given by Grisvard and the mixed formulation.