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 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.
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Abbreviations
- 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(Λk(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
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Hiptmair, R., Kotiuga, P.R. & Tordeux, S. Self-adjoint curl operators. Annali di Matematica 191, 431–457 (2012). https://doi.org/10.1007/s10231-011-0189-y
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DOI: https://doi.org/10.1007/s10231-011-0189-y
Keywords
- curl operators
- Self-adjoint extension
- Complex symplectic space
- Glazman-Krein-Naimark theorem
- Co-homology spaces
- Spectral properties of curl