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Self-adjoint curl operators

by R. Hiptmair and P. Kotiuga and S. Tordeux

(Report number 2008-27)

Abstract
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 extension. 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.

Keywords:

@Techreport{HKT08_37,
  author = {R. Hiptmair and P. Kotiuga and S. Tordeux},
  title = {Self-adjoint curl operators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-27},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-27.pdf },
  year = {2008}
}

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