1998
DOI: 10.1007/pl00004656
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On a selfadjoint realization of curl in exterior domains

Abstract: The first order vectorial operator 'curl' occurs in many areas of applications, e.g. in electrodynamics, fluid mechanics (hydrodynamics), magnetohydrodynamics and elasticity theory. It is the purpose of this note to discuss a particular instance of 'curl' where a domain can be given to 'curl' so that it becomes a selfadjoint operator. The particular domain of selfadjointness under consideration is singled out by its relevance in applications. To establish this operator we initially consider 'curl' as an operat… Show more

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Cited by 26 publications
(32 citation statements)
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“…Let us recall some results from [2]: • A is a selfadjoint operator with a compact resolvent operator in space L 2 σ . (Note that the selfadjointness of A in L 2 σ was also proved by Z. Yosida and Y. Giga in [26] and by R. Picard in [22].) Operator A commutes with projection P σ in space D 1 .…”
Section: A Navier-stokes Approximation Of a Solution Of The Euler Promentioning
confidence: 97%
“…Let us recall some results from [2]: • A is a selfadjoint operator with a compact resolvent operator in space L 2 σ . (Note that the selfadjointness of A in L 2 σ was also proved by Z. Yosida and Y. Giga in [26] and by R. Picard in [22].) Operator A commutes with projection P σ in space D 1 .…”
Section: A Navier-stokes Approximation Of a Solution Of The Euler Promentioning
confidence: 97%
“…This has in essence already been observed in [14] for simply connected domains with smooth boundary. We shall extend this result to fairly arbitrary open sets with a boundary permitting a suitable local compact embedding result, [15][16][17]. As a by-product, we can discuss well-posedness for a class of bi-anisotropic materials with "modified" Drude-Born-Fedorov type material behavior; some cases of which have found previous attention, compare [18,19].…”
Section: Introductionmentioning
confidence: 99%
“…is the usual weak derivative in L 2 (Ω), see [3] for the conceptual details. Containment of a field E in D c url is the proper weak generalization of the classical boundary condition "n × E = 0 on ∂Ω", whereas E ∈ D d iv generalized the classical boundary condition "n · E = 0 on ∂Ω".…”
mentioning
confidence: 99%
“…We shall denote the operator curl 776 Section 23: Applied operator theory subject to boundary conditions (1) by curl. Under fairly general assumptions it can be shown that curl is actually selfadjoint and even has -apart from 0 -only discrete spectrum σ d curl , for a first result see [3]. The most general result, [1], merely requires Ω to be an open set with bounded measure to obtain the same properties for curl.…”
mentioning
confidence: 99%