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

##### MSC:

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 |

##### Keywords:

extension; Maxwell’s operator; Lipschitz boundary; self-adjoint; orthogonal Weil expansions; Dirichlet problem; Neumann problem; Poisson equation; spectrum
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\textit{M. Sh. Birman} and \textit{M. Z. Solomyak}, Sib. Math. J. 28, No. 1--2, 12--24 (1987; Zbl 0655.35067); translation from Sib. Mat. Zh. 28, No. 1(161), 23--36 (1987)

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##### References:

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