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\(L_ 2\)-theory of the Maxwell operator in arbitrary domains. (English. Russian original) Zbl 0653.35075

Russ. Math. Surv. 42, No. 6, 75-96 (1987); translation from Usp. Mat. Nauk 42, No. 6, 61-76 (1987).
The problems of the precise definition and the description of the properties of the Maxwell operator corresponding to an ideal conducting boundary are discussed. Let \(\Omega\) be the domain in \({\mathbb{R}}^ 3\). The Maxwell operator acts according to the rule \[ M\{u^ 1,u^ 2\}=\{i\epsilon^{-1}rot u^ 2,-i\mu^{-1}rot u^ 1\}, \] under the conditions \[ div(\epsilon u^ 1)=div(\mu u^ 2)=0,\quad \gamma_{\tau}u^ 1=0,\quad \gamma_{\nu}(\mu u^ 2)=0. \] Here \(u^ 1\), \(u^ 2\) are the electric and magnetic vectors, the matrices \(\epsilon\), \(\mu\) give the dielectric and magnetic permeability, \(\gamma_{\nu}\), \(\gamma_{\tau}\) denote the normal and the tangential components of the vector field u on \(\partial \Omega.\)
The main results are as follows.
1. The suitable self-adjoint realization of M is described.
2. The self-adjoint elliptic operator \({\mathcal L}\) (8\(\times 8\)-system with some boundary conditions) is constructed such that this realization \({\mathfrak M}(\epsilon,\mu)\) is the part of \({\mathcal L}\) on some reducing subspace.
This fact permits one to use the theory of elliptic boundary value problems in studying the differential properties of the vector fields from Dom \({\mathfrak M}(\epsilon,\mu)\). Let s be a measurable, self-adjoint and positively defined matrix-function \(\Omega\), with real-valued entries, and let E(s,\(\tau)\), E(s,\(\nu)\) denote the spaces of gradients of the weak solutions of the equation \(div(s\phi)=f\) \((\in L_ 2)\), under the boundary condition \(\phi |_{\partial \Omega}=0\) (\(\tau\)- case) or \(\gamma_{\nu}(s\phi)=0\) (\(\nu\)-case).
3. For bounded \(\Omega\) with piecewise-smooth boundary and for \(\epsilon,\mu \in C^ 1({\bar \Omega})\) it is proved that if \(\{u^ 1,u^ 2\}\in {\mathcal D}om {\mathfrak M}(\epsilon,\mu)\) then \[ u^ 1\in H^ 1(\Omega)\quad (mod E(\epsilon,\tau)),\quad u^ 2\in H^ 1(\Omega)(mod E(\mu,\nu)). \] 4. For bounded \(\Omega\) with Lipschitz boundary it is proved that the spectrum of \({\mathfrak M}(\epsilon,\mu)\) is discrete.
Note that there is a misprint in the English translation: in Theorem 2.3 must be “bounded” instead of “unbounded”.
Reviewer: M.Solomyak

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
78A25 Electromagnetic theory (general)
35D05 Existence of generalized solutions of PDE (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35P05 General topics in linear spectral theory for PDEs
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