, Volume 166, Issue 1, pp 11-22
Date: 02 Mar 2010

On a partially isometric transform of divergence-free vector fields

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The paper deals with the so-called M-transform, which maps divergence-free vector fields in Ω T := {x ∈ Ω| dist(x, ∂Ω) < T}, Ω ⊂⊂ \( \mathbb{R} \) 3, to the space of transversal fields. The latter space consists of vector fields in Ω T tangential to the equidistant surfaces of the boundary ∂Ω. 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 < T ω, where T ω depends on the geometry of Ω. This paper provides a generalization for arbitrary T. It is proved that M T is partially isometric, and its intertwining properties are established. Bibliography: 6 titles.

Dedicated to N. N. Uraltseva on the occasion of her jubilee
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 370, 2009, pp. 22–43.