Self-adjoint curl operators

Hiptmair, Ralf; Kotiuga, Peter; Tordeux, Sébastien

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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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Titre

Self-adjoint curl operators

Auteur

Hiptmair, Ralf. Seminar for Applied Mathematics, ETH Zurich, 8092, Zurich, Switzerland
Kotiuga, Peter. Department of Electrical and Computer Engineering, Boston University, Boston, MA, USA
Tordeux, Sébastien. Institut de Mathématiques de Toulouse, INSA-Toulouse, Toulouse, France

Type de document

Postprint

Langue

Anglais

Publié dans

Annali di Matematica Pura ed Applicata, 2012, vol. 191, no. 3, p. 431-457. Springer-Verlag

Autre version électronique

Publisher's version : https://doi.org/10.1007/s10231-011-0189-y

Classification

Mathématiques

Mots clés

curl operators ; Self-adjoint extension ; Complex symplectic space ; Glazman-Krein-Naimark theorem ; Co-homology spaces ; Spectral properties of curl ; C ∞( D ) : Infinite differentiable functions on D, C ∞(D)=(C ∞(D))3 ; $${C^{\infty}_{0}(D)}$$ : Compactly supported functions in C ∞(D), $${{\rm {\bf C}}_{0}^{\infty}(D)=(C^{\infty}_{0}(D))^{3}}$$ ; $${{\rm curl}_{\partial}}$$ : Scalar valued surface rotation ; $${{\mathsf{d}}}$$ : Exterior derivative of differential forms ; $${{\partial}{M}}$$ : Boundary of M ; $${{\mathcal D}(\mathsf{T})}$$ : Domain of definition of the linear operator $${\mathsf{T}}$$ ; D : Bounded (open) Lipschitz domain in affine space $${{\mathbb R}^{3}}$$ ; D ′ : Closure of the complement of D, $${D':=\mathbb{R}^{3}{\setminus}\bar{D}}$$ ; $${{\rm div}_\partial}$$ : Surface divergence ; $${{\bf grad}_\partial}$$ : Surface gradient ; H ( curl , D ) : Real Hilbert space $${\{{\rm {\bf v}}\in {\varvec L}^2: \,{\rm {\bf curl}}\,{\rm {\bf v}}\in {\bf L}^2\}}$$ with graph norm ; H 0( curl , D ) : Closure of $${{\rm {\bf C}}_{0}^{\infty}(D)}$$ in H(curl, D) ; $${H^{\frac{1}{2}} (\partial{D})}$$ : Trace space of $${H^1(D):=\{u\in {L^2(D)}:\nabla {u} \in {L^2(D)}\}}$$ ; $${\mathbf{H}^{-\frac{1}{2}}_{\mathbf{t}}({\rm curl}_\partial, \partial{D})}$$ : Tangential traces of vector fields in H(curl, D) ; $${\mathbf{H}^{s}_{\mathbf{t}}(\partial{D}),\mathbf{L}^{2}_{\mathbf{t}}(\partial{D})}$$ : Tangential trace spaces ; $${H{\frac{3}{2}}(\partial{D})}$$ : See (5.3) ; $${HF^{k}({\mathsf{d}},D)}$$ : Square integrable k-forms with square integrable exterior derivative ; $${HF^{k}_{0}({\mathsf{d}},D)}$$ : Completion of compactly supported k-forms in $${HF^{k}({\mathsf{d}},D)}$$ ; $${HF^{-\frac{1}{2},k}({\mathsf{d}}, {\partial}{D})}$$ : Trace space of $${HF^{k}({\mathsf{d}},D)}$$ ; $${HZ^{-\frac{1}{2},k}({\partial}{D})}$$ : Closed k-forms in $${{HF^{-\frac{1}{2},1}({\mathsf{d}}, {\partial}{D})}}$$ , see for instance (6.1) ; $${HF^{\frac{3}{2},0}({\partial}{D})}$$ : See (5.5) ; $${{\mathcal H}^{1}({\partial}{D})}$$ : Co-homology space of harmonic 1-forms on $${{\partial}{D}}$$ ; i * : Natural trace operator for differential forms ; L 2( D ) : Real square integrable functions on D, L 2(D)=(L 2(D))3 ; $${{L}^{\sharp}}$$ : Symplectic orthogonal of subspace L of a symplectic space ; L 2(Λ k ( M )) : Hilbert space of square integrable k-forms on M ; n : Exterior unit normal vector field on $${{\partial}{D}}$$ ; $${S_i,S^{\prime}_{i}}$$ : Inside and outside cuts of D, see Sect. 6.3 ; $${{\mathcal N}({\mathsf{T}})}$$ : Kernel (null space) of linear operator $${\mathsf{T}}$$ ; $${{\mathcal R}({\mathsf{T}})}$$ : Range space of a linear operator $${\mathsf{T}}$$ ; $${\mathsf{T},\mathsf{T}*}$$ : An (unbounded) linear operator and its adjoint ; $${\mathsf{T}_{min}}$$ , $${\mathsf{T}_{s}}, \mathsf{T}_{max}$$ : Min, self-adj. and max closures of a symmetric operator $${\mathsf{T}}$$ ; $${{\rm {\bf u}},{\rm {\bf v}}, \ldots}$$ : Vector fields on a three-dimensional domain or elements of trace space of vector proxies ; γ t , γ n : Tangential and normal boundary traces of a vector field ; $${\omega,\eta,\ldots}$$ : Differential forms ; $${\omega^{0},\omega^{\perp}}$$ : Components of the Hodge decomposition of $${\omega \in {HF^{-\frac{1}{2},1}}{(\mathsf{d},{\partial}{D})}}$$ ; $${\langle\cdot\rangle}$$ : (Relative) Homology class of a cycle ; $${\wedge}$$ : Exterior product of differential forms ; $${\star}$$ $$({\star_{g}})$$ : Hodge operator (induced by metric g) ; · : Euclidean inner product in $${{\mathbb R}^{3}}$$ ; × : Cross product of vectors $${\in{\mathbb {R}}^{3}}$$ ; (·, ·) : Inner product: for $${\omega\in L^{2}(\Lambda^{k}(M))}$$ , $${(\omega,\omega)_{k,M}=\int_{M}\omega\wedge\star\omega}$$ ; ||·|| : Norm: for $${\omega\in L^{2}(\Lambda^{k}(M))}$$ , $${\|\omega\|_{k,M}^{2} := (\omega,\omega)_{k,M}}$$ ; [·, ·] : Symplectic pairing: for 1-forms on 2-manifold M, $${\left[\omega,\eta\right]_{M}=\int_{M} \omega\wedge\eta}$$ ; [·]Γ : Jump of trace of a function across 2-manifold Γ

Identifiant OAI-PMH

oai:doc.rero.ch:311991

Summary

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

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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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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