Hiptmair, RalfKotiuga, Peter RobertTordeux, Sebastien2018-07-192018-07-192012-09-01Ralf Hiptmair, Peter Robert Kotiuga, Sebastien Tordeux. 2012. "Self-adjoint curl operators." Annali di Matematica Pura ed Applicata, Volume 191, Issue 3, pp. 431 - 457 (27). doi:10.1007/s10231-011-0189-y0373-3114https://hdl.handle.net/2144/29997This is a post-peer-review, pre-copyedit version of an article published in Annali di Matematica Pura ed Applicata. The final published version is available online at: http://dx.doi.org/10.1007/s10231-011-0189-y”.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 ∧-product of 1-forms on ∂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.p. 431 - 457Science & technologyPhysical sciencesMathematics, appliedMathematicsCurl operatorsSelf-adjoint extensionComplex symplectic spaceGlazman-Krein-Naimark theoremCo-homology spacesSpectral properties of curlFree magnetic fieldsCompact embeddingHodge decompositionsLipschitz polyhedraPure mathematicsGeneral mathematicsArticle10.1007/s10231-011-0189-y