Complex Friedrichs systems and applications

Recently, there has been a significant development of the abstract theory of Friedrichs systems in Hilbert spaces (Ern, Guermond & Caplain, 2007; Antoni´c & Burazin, 2010), and its applications to specific problems in mathematical physics. However, these applications were essentially restricted to real systems. We check that the already developed theory of abstract Friedrichs systems can be adjusted to the complex setting, with some necessary modifications, which allows for applications to partial differential equations with complex coefficients. We also provide examples where the involved Hilbert space is not the space of square integrable functions, as it was the case in previous works, but rather its closed subspace or the space Hs(Rd;Cr), for real s. This setting appears to be suitable for particular systems of partial differential equations, such as the Dirac system, the Dirac-Klein-Gordon system, the Dirac-Maxwell system, and the time-harmonic Maxwell system, which are all addressed in the paper. Moreover, for the time-harmonic Maxwell system we also applied a suitable version of the two-field theory with partial coercivity assumption which is developed in the paper.
symmetric positive first-order system, two-field theory, partial differential equations with complex coeffcients, coupled systems of partial differential equations