Journal of Mathematical Sciences

, Volume 206, Issue 3, pp 247–259 | Cite as

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.


Inverse Problem Riemannian Manifold Steklov Institute Adjoint Operator Integral Identity 
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  1. 1.
    M. I. Belishev, V. M. Isakov, L. N. Pestov, and V. A. Sharafutdinov, “On the Reconstruction of Metrics via External Electromagnetic Measurements,” Dokl. Math., 61, No. 3, 353–356 (2000).MathSciNetGoogle Scholar
  2. 2.
    M. I. Belishev and A. K. Glasman, “Dynamical inverse problem for the Maxwell system: recovering the velocity in a regular zone (the BC–method),” St. Petersb. Math. Journal, 12, No. 2, 279–316, (2001).MathSciNetGoogle Scholar
  3. 3.
    M. I. Belishev, “On a unitary transform in the space L 2(Ω;ℝ3) connected with the Weyl decomposition,” J. Math. Sci., 117, No. 2, 3900–3909 (2003).CrossRefMathSciNetGoogle Scholar
  4. 4.
    M. N. Demchenko, “On a partially isometric transform of divergence-free vector fields,” J. Math. Sci., 166, No. 1, 11–22 (2010).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    M. N. Demchenko, “The dynamical 3-dimensional inverse problem for the Maxwell system,” St. Petersb. Math. Journal, 23, 943–975 (2012).CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    G. Schwarz, “Hodge decomposition – a method for solving boundary value problems,” Lecture Notes Math., 1607 (1995).Google Scholar
  7. 7.
    T. Kato, Perturbation Theory For Linear Operators, Springer-Verlag, Berlin (1966).CrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.St. Petersburg Department of the V. A. Steklov Institute of Mathematics of the Russian Academy of SciencesSt. PetersburgRussia

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