zbMATH — the first resource for mathematics

Maxwell operator in regions with nonsmooth boundaries. (English. Russian original) Zbl 0655.35067
Sib. Math. J. 28, No. 1-2, 12-24 (1987); translation from Sib. Mat. Zh. 28, No. 1(161), 23-36 (1987).
The orthogonal extension of Maxwell’s operator is considered. In the region \(\Omega\) with Lipschitz boundary such an operator \({\mathcal L}\) turns out to be self-adjoint and can be reduced to orthogonal Weil expansions. It turns out that the functions from \(\text{Dom}\,{\mathcal L}\) to terms in \(H'(\Omega)\) are gradients of the weak solutions of the Dirichlet and Neumann problems for the Poisson equation. This permits to transfer automatically the information about the behaviour of such solutions depending on the properties of the boundary to the Maxwell operator. A noteworthy result is that the spectrum of the operator \({\mathcal L}\) in regions with Lipschitz boundary is discrete.
Reviewer: A.Borisov

35Q61 Maxwell equations
78A25 Electromagnetic theory, general
35J25 Boundary value problems for second-order elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI
[1] É. B. Bykhovskii and N. V. Smirnov, ?Orthogonal decomposition of the space of vector functions square integrable in a given region and vector analysis operators,? Trans. of the V. A. Steklov Inst. of Math., USSR Academy of Sciences,59, 5 (1960).
[2] M. Sh. Birman, ?Maxwell operator in regions with edges,? J. Sov. Math.,37, No. 1 (1987).
[3] M. Sh. Birman, ?Maxwell operator for a resonator with reentrant edges,? Vestn. Leningr. Gos. Univ., Ser. 1, No. 3, 3 (1986).
[4] I. S. Gudovich and S. G. Krein, ?Boundary-value problems for overdetermined systems of partial differential equations,? in: Differential Equations [in Russian], No. 9, Vilnius (1974), pp. 1-146.
[5] V. A. Kondrat’ev and O. A. Oleinik, ?Boundary-value problems for partial differential equations in nonsmooth regions,? Usp. Mat. Nauk,38, 3 (1983).
[6] V. Zaionchkovskii and V. A. Solonnikov, ?Neumann problem for second-order elliptic equations with edges on a boundary,? J. Sov. Math.,27, No. 2, (1984).
[7] V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad State Univ. (1985).
[8] O. A. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordan and Breach (1969). · Zbl 0184.52603
[9] O. A. Ladyzhenskaya and V. A. Solonnikov, ?Problems of vector analysis and generalized formulations of boundary-value problems for Navier-Stokes equations,? J. Sov. Math.,10, No. 2 (1987). · Zbl 0346.35084
[10] V. A. Kondrat’ev, ?Boundary-value problems for elliptic equations in regions with conical or angular points,? Tr. Mosk. Mat. Ob.,16, 209 (1967).
[11] V. A. Kondrat’ev, ?Smoothness of the solution of the Dirichlet problem for a secondorder elliptic equation in the neighborhood of an edge,? Diff. Uravn.,6, 1831 (1970).
[12] V. A. Kondrat’ev, ?Aspects of the solution of the Dirichlet problem for second-order elliptic equations in the neighborhood of an edge,? Diff. Uravn.,13, 2026 (1977).
[13] V. G. Maz’ya and B. A. Plamenevskii, ?First boundary-value problem for classical equations of mathematical physics in regions with piecewise smooth boundaries. I, II,? Z. Anal. Anwendungen,2, 335, 523 (1983).
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.