# zbMATH — the first resource for mathematics

Remarks on spectra of operator rot. (English) Zbl 0676.47012
Spectral properties of the operator rot have been studied in various function spaces of vector fields on a three dimensional domain. A self- adjoint operator associated with rot has been introduced by choosing appropriate boundary conditions. Eigenfunctions of the self-adjoint rot operator span the orthogonal complement of the irrotational fields in the Lebesgue space $$L^ 2(\Omega)$$ of square integrable vector fields. Eigenvalue problems for rot have been also discussed under different boundary conditions.
Reviewer: Z.Yoshida

##### MSC:
 47B25 Linear symmetric and selfadjoint operators (unbounded) 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
Full Text:
##### References:
 [1] Borchers, W., Sohr, H.: The equations divu=f and rotv=g with zero boundary conditions. Hokkaido Math. J.19, 67–87 (1990) · Zbl 0719.35014 [2] Bourguignon, J.P., Brezis, H.: Remarks on the Euler equation. J. Funct. Anal.15, 341–363 (1974) · Zbl 0279.58005 · doi:10.1016/0022-1236(74)90027-5 [3] Chandrasekhar, S., Kendall, P.C.: On force-free magnetic fields. Astrophys. J.126, 457–460 (1957) · doi:10.1086/146413 [4] Duvaut, G., Lions, J.L.: Les inequations en mecanique et en physique. Paris: Dunod 1972 · Zbl 0298.73001 [5] Foias, C., Temam, R.: Remarques sur les equations de Navier-Stokes stationaires et les phenomenes successifs de bifurcation. Ann. Sc. Norm. Super Pisa5, 29–63 (1978) · Zbl 0384.35047 [6] Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications I. Berlin Heidelberg New York: Springer 1972 · Zbl 0223.35039 [7] Montgomery, D., Turner, L., Vahala, G.: Three-dimensional magnetohydrodynamic turbulence in cylindrical geometry. Phys. Fluids21, 757–764 (1978) · Zbl 0376.76080 · doi:10.1063/1.862295 [8] Morrey, C.B.: Multiple integrals in the calculus of variations. Berlin Heidelberg New York: Springer 1966 · Zbl 0142.38701 [9] Taylor, J.B.: Relaxation of toroidal plasma and generation of reversed magnetic fields. Phys. Rev. Lett.33, 1139–1141 (1974) · doi:10.1103/PhysRevLett.33.1139
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.