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A Riemann-Hilbert Problem for the Moisil-Teodorescu System

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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Correspondence to A. N. Polkovnikov.

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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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Keywords

  • Dirac operator
  • Riemann-Hilbert problem
  • Fredholm operators