The admissible boundary conditions for symmetric positive systems of first-order linear partial differential equations, originally introduced by Friedrichs [11], were recently related to three different sets of intrinsic geometric conditions in graph spaces [10]. We rewrite their cone formalism in terms of an indefinite inner product space, which in a quotient by its isotropic part gives a Kreȋn space. This new viewpoint allows us to show that the three sets of intrinsic boundary conditions are actually equivalent, which will hopefully facilitate further investigation of their precise relation to the original Friedrichs boundary conditions.

We revisit homogenisation theory for Friedrichs systems. In particular, we show that G-compactness can be obtained under severely weaker assumptions than in the original work of Burazin and Vrdoljak (2014). In this way we extend the applicability of G-compactness results for Friedrichs systems to equations that yield memory effects in the homogenised limit and detour any usage of compactness techniques previously employed.

No abstract

Recently, there has been a significant development of the abstract theory of Friedrichs systems in Hilbert spaces [Ern, A., Guermond, J.-L., and Caplain, G., Commun. Partial Differ. Equations 32, 317–341 (2007) and Antonić, N. and Burazin, K., Commun. Partial Differ. Equations 35, 1690–1715 (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 studies, 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 this paper.

MSC:Keywords: Symmetric positive system First-order system of partial differential equations Boundary condition Boundary operator Following the recent progress in understanding the abstract setting for Friedrichs symmetric positive systems by Ern, Guermond and Caplain (2007) [8], as well as Antonić and Burazin (2010) [3], an attempt is made to relate these results to the classical Friedrichs theory. A comparison of two approaches, via the trace operator and the boundary operator, has been made, favouring the latter. Finally, a particular set of sufficient conditions for a boundary matrix field to define a boundary operator in that case is given, and the applicability of this procedure in realistic situations is shown by examples.