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/ . The series converges if the Lipschitz constant of the perturbation is small.