Intrinsic Boundary Conditions for Friedrichs Systems

Abstract

The admissible boundary conditions for symmetric positive systems of first-order linear partial differential equations, originally introduced by Friedrichs [11 Friedrichs , K.O. ( 1958 ). Symmetric positive linear differential equations . Comm. Pure Appl. Math. 11 : 333 – 418 .[Crossref], [Web of Science ®] , [Google Scholar]], were recently related to three different sets of intrinsic geometric conditions in graph spaces [10 Ern , A. , Guermond , J.-L. , Caplain , G. (2007). An intrinsic criterion for the bijectivity of Hilbert operators related to Friedrichs’ systems. Comm. Part. Diff. Eqs. 32:317–341.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. We rewrite their cone formalism in terms of an indefinite inner product space, which in a quotient by its isotropic part gives a Kreǐn 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.