Symmetric positive systems of 1st order linear PDE were introduced by Kurt Otto Friedrichs in 1958 (whence "Friedrichs systems") in an attempt at treating equations that change their type, like the equations modelling a transonic fluid flow. A Friedrichs system consists of a certain 1st order system of PDE and an admissible boundary condition. Friedrichs showed that this class of problems encompasses a wide variety of classical and neo-classical initial and boundary value problems (IVP/BVP) for various linear PDE, such as BVP for some elliptic systems, the Cauchy problem for linear symmetric hyperbolic systems, mixed IVP/BVP for hyperbolic equations, BVP for equations of mixed type like the Tricomi equation. Friedrichs' main goal was to derive systematically the type of conditions that need be imposed on various parts of the boundary, as was done by Cathleen Synge Morawetz for the Tricomi equation.
Inclusion of such a variety of different problems, each with technical peculiarities of its own, into a single framework requires all different features to be included as well -- a challenge which many authors have worked on, including Morawetz, Lax, Rauch, Phillips, and Sarason. More recently, Ern, Guermond, and Caplain (2007), inspired by their interest in the numerical treatment of Friedrichs systems, suggested another approach: to express a Friedrichs system in terms of operators acting on abstract Hilbert spaces and to prove well-posedness results in this abstract setting. We (Antonić & Burazin, 2010) re-wrote their cone formalism in terms of an indefinite inner product space, which in a quotient by its isotropic part gives a Krein space. This new view-point allowed us to show the equivalence of the three sets of intrinsic boundary conditions.
Most classical papers deal with Friedrichs systems in real space setting. In this talk we shall address the extension of both stationary and non-stationary theory to complex spaces, as well as the two-field theory, commenting on the difficulties encountered in the semi-linear case, as well as in the Banach space setting. Finally, we shall discuss the applicability of these extensions to some examples, like the Dirac or Maxwell system.