Abstract
Let Ω be a bounded domain inR 3 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 {H 1(Ω)}3 ∩V(Ω) when Ω is convex, whereH 1(Ω) is the usual first order Sobolev space. To this end, we adopt the techniques given by Grisvard and the mixed formulation.
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Kikuchi, F. An isomorphic property of two Hilbert spaces appearing in electromagnetism: Analysis by the mixed formulation. Japan J. Appl. Math. 3, 53–58 (1986). https://doi.org/10.1007/BF03167091
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DOI: https://doi.org/10.1007/BF03167091