Geodesic
On a partially isometric transform of divergence free vector fields
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 22-43 
M. N. Demchenko. On a partially isometric transform of divergence free vector fields. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 22-43. http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a1/
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     author = {M. N. Demchenko},
     title = {On a partially isometric transform of divergence free vector fields},
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     year = {2009},
     volume = {370},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a1/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The paper deals with the so-called $M$-transform which maps divergence free vector fields in $\Omega^T:=\{x\in\Omega\mid\operatorname{dist}(x,\partial\Omega), $\Omega\subset\subset\mathbb R^3$, to the space of transversal fields. The latter space consists of the vector fields in $\Omega^T$ tangential to the equidistant surfaces of boundary $\partial\Omega$. In papers devoted to the dynamical inverse problem for the Maxwell system, in the framework of the BC-method, the operator $M^T$ was defined for $T, where $T_\omega$ depends on the geometry of $\Omega$. This paper provides the generalization for arbitrary $T$. It is proved that $M^T$ is partially isometric and its intertwining properties are established. Bibl. – 6 titles.

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[2] M. I. Belishev, A. K. Glasman, “Dinamicheskaya obratnaya zadacha dlya sistemy Maksvella: vosstanovlenie skorosti v regulyarnoi zone (BC-metod)”, Algebra i analiz, 12:2 (2000), 131–187 | MR | Zbl

[3] M. I. Belishev, “Ob unitarnom preobrazovanii v prostranstve $L_2(\Omega;\mathbb R^3)$, svyazannom s razlozheniem Veilya”, Zap. nauchn. semin. POMI, 275, 2001, 25–40 | MR | Zbl

[4] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975 | MR | Zbl

[5] H. Sohr, The Navier–Stokes Equations: an Elementary Functional Analytic Approach, Birkhäuser, Berlin, 2001 | MR | Zbl

[6] T. Kato, Teoriya vozmuschenii lineinykh operatorov, M., 1972