Self-adjoint curl operators
Hiptmair, Ralf ; Kotiuga, Peter ; Tordeux, Sébastien
In: Annali di Matematica Pura ed Applicata, 2012, vol. 191, no. 3, p. 431-457
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- We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain D. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem, this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl, D) equipped with a symplectic pairing arising from the $${\wedge}$$ -product of 1-forms on $${\partial D}$$ . Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extensions. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed