Browsing by Author "Erceg, Marko"

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    Complex Friedrichs systems and applications
    (2017-01) Antonić, Nenad; Burazin, Krešimir; Crnjac, Ivana; Erceg, Marko; Mathematics
    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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    Friedrichs systems in a Hilbert space framework: solvability and multiplicity
    (2017-04) Antonić, Nenad; Erceg, Marko; Michelangeli, Alessandro; Mathematics
    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 Antonić 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 suffcient 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.