On a partially isometric transform of divergence free vector fields

M. N. Demchenko

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 22-43

Résumé

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.