A method for solving ill-posed nonlocal problem for the elliptic equation with data on the whole boundary

Abstract

In this paper a nonlocal problem for the elliptic equation in a cylindrical domain is considered. It is shown that this problem is ill-posed as well as the Cauchy problem for the Laplace equation. The method of spectral expansion in eigenfunctions of the nonlocal problem for equations with involution establishes a criterion of the strong solvability of the considered nonlocal problem. It is shown that the ill-posedness of the nonlocal problem is equivalent to the existence of an isolated point of the continuous spectrum for a nonself-adjoint operator with involution.

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Correspondence to Berikbol T. Torebek.

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Kal’menov, T.S., Torebek, B.T. A method for solving ill-posed nonlocal problem for the elliptic equation with data on the whole boundary. J. Pseudo-Differ. Oper. Appl. 10, 177–185 (2019). https://doi.org/10.1007/s11868-017-0231-y

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Keywords

  • Elliptic operator
  • Nonlocal boundary conditions
  • Operator with involution
  • Criterion of well-posedness
  • Riesz basis

Mathematics Subject Classification

  • Primary 35J25
  • 35C10
  • Secondary 35P10