Self-adjoint curl operators

Ralf Hiptmair (1) , P. Robert Kotiuga (2) , Sébastien Tordeux (3)

1 ETH-SAM - Seminar for Applied Mathematics
2 ECE - Department of Electrical and Computer Engineering [Boston University]
3 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Mots clés: curl operator; self-adjoint extension; complex symplectic space; Glazman-Krein- Naimark theorem; co-homology spaces; spectral properties of curl AMS subject classifications 47F05; 46N20

Domaines: Analyse fonctionnelle [math.FA]

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