On a partially isometric transform of divergence-free vector fields

Abstract

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.