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Decomposition of \(H_{div}^{-1/2}(\Gamma )\) and nature of Steklov-Poincaré operator of exterior electromagnetism problem. (Décomposition de \(H_{div}^{-1/2}(\Gamma )\) et nature de l’opérateur de Steklov-Poincaré du problème extérieur de l’électromagnétisme.) (French) Zbl 0767.35094

Summary: We show that, if \(\Gamma\) is diffeomorphic to \(S^ 2\), then \(H_{\text{div}}^{-1/2}(\Gamma)\) is splitted into \[ \nabla_ \Gamma (H^{3/2}(\Gamma))_ \oplus^ \perp\text{ rot}_ \Gamma(H^{1/2}(\Gamma)). \] Then we show that the Steklov-Poincaré operator for the exterior electromagnetism problem, which maps the tangential component \(E\wedge n\) of the electric field to the electric current \(j\) is the direct sum of an operator going from \(\nabla_ \Gamma(H^{3/2}(\Gamma))\) to \(\text{rot}_ \Gamma(H^{1/2}(\Gamma))\) and an operator going from \(\text{rot}_ \Gamma(H^{1/2}(\Gamma))\) to \(\nabla_ \Gamma(H^{3/2}(\Gamma))\) modulo a regularizing operator.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
78A45 Diffraction, scattering
31A10 Integral representations, integral operators, integral equations methods in two dimensions
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