On Transforms of Divergence-Free and Curl-Free Fields, Associated with Inverse Problems
The M- and N-transforms acting, respectively, on divergence-free and curl-free vector fields on a Riemannian manifold with boundary are investigated. These transforms arise in studying the inverse problems of electrodynamics and elasticity theory. A divergence-free field is mapped by M to a field that is tangential to equidistants of the boundary. The N-transform maps a curl-free field to a field that is normal to equidistants. In preceding papers, the operators M and N were considered in the case of smooth equidistants, which is realized in a sufficiently small near-boundary layer. This allows one to consider transforms of fields supported in such a layer; it was proved that M and N are unitary in the corresponding spaces with L2-norms. In one of the papers, the case of fields on the whole manifold was considered, but almost all equidistants were assumed to be Lipschitz surfaces. It was proved that M is coisometric (i.e., the adjoint operator is isometric). In the present paper, the same result is obtained for both transforms in the general case with no constraints on equidistants at all.
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