FAIERMAN, M and MENNICKEN, R and MOLLER, M (1995) A BOUNDARY EIGENVALUE PROBLEM FOR A SYSTEM OF PARTIAL-DIFFERENTIAL OPERATORS OCCURRING IN MAGNETOHYDRODYNAMICS. MATHEMATISCHE NACHRICHTEN, 173. pp. 141-167. ISSN 0025-584X,
Full text not available from this repository.Abstract
In the recent paper [ALMS] a method was proposed for dealing with the spectral theory for pencils of the form L(0) - mu M, where L(0) and M are 2 x 2 block operator matrices acting in a Banach space and M is invertible. The authors then applied their techniques to a problem occurring in magnetohydrodynamics (see [L]) wherein the entries of L(0) were ordinary differential operators and M was the identity operator, and were completely able to characterize the essential spectrum of the operator L(0). In the same book [L], the author, A. E. LIFSCHITZ, also considers a problem wherein the entries of L(0) are partial differential operators acting on a rectangular domain in R(2) with coefficients having singularities at one edge of the boundary, and some properties of the essential spectrum of L(0) are proved. However, it appears that a complete mathematical treatment of this problem has not yet been undertaken. The object of this paper is to show that when the operators involved are all regular, then the spectral theory for the pencil L(0) - mu M can be completely dealt with using the methods of [ALMS], and indeed we shall give a complete characterization of the essential spectrum of the pencil for this case. We might add that the problem of LIFSCHITZ under consideration here was also considered by DESCLOUX and GEYMONAT in [DG1, DG2]. However, their approach differs from ours in that they work completely with quadratic forms. Our approach is strictly operator-theoretic and is an attempt to reinforce the quite general feeling that the method proposed in [ALMS] is precisely the correct one for dealing with many of the problems occurring in magnetohydrodynamics. Finally, we refer the reader to HAMEIRI [H] and KAKO [K] for further discussions concerning this problem. The regular spectral problem treated in the present paper concerns MHD oscillations of so-called hard core equilibrium configurations where the plasma is confined between two concentric toroidal rigid perfectly conducting walls. The physically more relevant equilibrium configurations where the plasma is inside one toroidal rigid perfectly conducting wall leads to a singular spectral problem of similar type which will be treated by the authors in a subsequent article.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | ESSENTIAL SPECTRUM; |
| Depositing User: | Dr. Gernot Deinzer |
| Last Modified: | 19 Oct 2022 08:38 |
| URI: | https://pred.uni-regensburg.de/id/eprint/52870 |
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