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) 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.
Type de document :
Article dans une revue
Annali di Matematica Pura ed Applicata, Springer Verlag, 2012, 191 (3), pp.431-457. 〈10.1007/s10231-011-0189-y〉

https://hal.inria.fr/inria-00527733
Contributeur : Sébastien Tordeux <>
Soumis le : mercredi 20 octobre 2010 - 10:47:05
Dernière modification le : mercredi 23 mai 2018 - 17:58:04

### Citation

Ralf Hiptmair, P. Robert Kotiuga, Sébastien Tordeux. Self-adjoint curl operators. Annali di Matematica Pura ed Applicata, Springer Verlag, 2012, 191 (3), pp.431-457. 〈10.1007/s10231-011-0189-y〉. 〈inria-00527733〉

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