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A Riemann-Hilbert Problem for the Moisil-Teodorescu System

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Abstract

In a bounded domain with smooth boundary in ℝ3 we consider the stationary Maxwell equations for a function u with values in ℝ3 subject to a nonhomogeneous condition (u, v)x = u0 on the boundary, where v is a given vector field and u0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.

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Authors and Affiliations

  1. Siberian Federal University, Institute of Mathematics and Computer Science, Krasnoyarsk, 660041, Russia

    A. N. Polkovnikov

  2. Institute of Mathematics, University of Potsdam, Potsdam, 14476, Germany

    N. Tarkhanov

Authors
  1. A. N. Polkovnikov
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  2. N. Tarkhanov
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Correspondence to A. N. Polkovnikov.

Additional information

Original Russian Text © A.N. Polkovnikov and N. Tarkhanov, 2018, published in Matematicheskie Trudy, 2018, Vol. 21, No. 1, pp. 155–192.

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Polkovnikov, A.N., Tarkhanov, N. A Riemann-Hilbert Problem for the Moisil-Teodorescu System. Sib. Adv. Math. 28, 207–232 (2018). https://doi.org/10.3103/S1055134418030057

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  • DOI: https://doi.org/10.3103/S1055134418030057

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