engleski

On equivalence of two descriptions of boundary conditions for Friedrichs systems

Friedrichs systems are a class of boundary value problems which allows the study of a wide range of differential equations in a unified framework. They were introduced by K. O. Friedrichs in 1958 in an attempt to treat equations of mixed type (such as the Tricomi equation). The Friedrichs system consists of a first order system of partial differential equations (of specific type) and an admissible boundary condition enforced by a matrix-valued boundary field. In a recent paper: A. Ern, J.-L. Guermond, G. Caplain: An Intrinsic Criterion for the Bijectivity Of Hilbert Operators Related to Friedrichs’ Systems, Comunications in Partial Differential Equations, 32, (2007), 317– 341. a new approach to the theory of Friedrichs systems has been proposed, rewritting them in terms of Hilbert spaces, and a new way of representing the boundary conditions has been introduced. The admissible boundary conditions are characterised by two intrinsic geometric conditions in the graph space, thus avoiding the traces at the boundary. Another representation of boundary conditions via boundary operators has been introduced as well, which is equivalent to the intrinsic one (those enforced by two geometric conditions) if the sum of two specific subspaces V and ˜ V of the graph space is closed. However, the validity of the last condition was left open. We have noted that these two geometric conditions can be naturally written in the terms of an indefinite inner product on the graph space, and the application of classical results on Krein spaces allowed us to construct a counter– example, which shows that V + ˜ V is not necessarily closed in the graph space. In the case of one space dimension we will give complete classification of admissible boundary conditions (those satisfying two geometric conditions). The relation between the classical representation of admissible boundary conditions (via matrix fields on the boundary), and those given by boundary operator will be addressed as well.

Friedrichs systems ; boundary conditions ; Krein space

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