Geodesic
On transforms of divergence-free and curl-free fields, connected with inverse problems
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 43, Tome 422 (2014), pp. 27-46 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study $M$- and $N$-transform acting correspondingly on divergence-free and curl-free vector fields on Riemannian manifold with boundary. These transforms arise in the study of inverse problems of electrodynamics and elasticity theory. A divergence-free field $y$ is mapped by $M$ to a field that is tangential to equidistants of the boundary. $N$-transform maps curl-free field to a field that is normal to equidistants. In preceding papers operators $M$ and $N$ were considered in case of smooth equidistants, which is realized in a small enough near-boundary layer. This allows to consider transforms of fields supported in such a layer; it was proved that $M$ and $N$ are unitary in corresponding spaces with $L_2$-norms. In one of the papers the case of fields on the whole manifold was considered, but almost all equidistants were supposed to be Lipschitz surfaces. It was proved that $M$ is coisometric (i.e., adjoint operator is isometric). In this paper, we obtain the same result for both transforms in the general case with no constraints on equidistants at all. 
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M. N. Demchenko. On transforms of divergence-free and curl-free fields, connected with inverse problems. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 43, Tome 422 (2014), pp. 27-46. http://geodesic.mathdoc.fr/item/ZNSL_2014_422_a2/

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