Friedrichs systems in a Hilbert space framework: Solvability and multiplicity

[摘要] The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonia and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems, and proving that any pair of abstract Friedrichs operators admits bijective extensions with a signed boundary map. Moreover, we provide sufficient and necessary conditions for existence of infinitely many such pairs of spaces, and by the universal operator extension theory (Grubb, 1968) we get a complete identification of all such pairs, which we illustrate on two concrete one-dimensional examples. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Symmetric positive first-order system of partial differential equations;Krein space;Universal parametrisation of extensions