Complex Friedrichs systems and applications
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Date
2017-01
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Abstract
Recently, there has been a significant development of the abstract theory of Friedrichs
systems in Hilbert spaces (Ern, Guermond & Caplain, 2007; Antoni´c & Burazin,
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
works, 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 the paper.
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Keywords
symmetric positive first-order system, two-field theory, partial differential equations with complex coeffcients, coupled systems of partial differential equations