Quadratic functional and integral equations for the harmonic Maxwell's equations in exterior domains

We exhibit a quadratic functional for the harmonic Maxwell equations in exterior domain. The functional reaches its unique minimum in the space of our-going and/or in-going solution at the out-going solutions with absorbing boundary condition. It gives a ''natural'' Hilbertian framework to this problem. It is possible to compute explicitly all the terms of the functional to obtain a mixed integral system. Simple algorithms solve the system.