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Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem

Abstract

In a special Lipschitz domain treated as a perturbation of the upper half-space, we construct a perturbation theory series for a positive harmonic function with zero trace. The terms of the series are harmonic extensions to the half-space from its boundary of distributions defined by a recurrent formula and passage to the limit. The approximation error by a segment of the series is estimated via a power of the seminorm of the perturbation in the homogeneous Slobodestkiĭ space b 1−1/N N . The series converges if the Lipschitz constant of the perturbation is small.

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Correspondence to A. I. Parfenov.

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Original Russian Text © A.I. Parfenov, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 1, pp. 158–200.

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Parfenov, A.I. Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem. Sib. Adv. Math. 27, 274–304 (2017). https://doi.org/10.3103/S1055134417040058

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Keywords

  • positive harmonic function
  • conformal mapping
  • Lipschitz continuous perturbation of the boundary