## 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