Solution of the Dirichlet Problem for a Schlicht Region

Part of the Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen book series (GL, volume 162)


As an essential tool in the following discussion we shall use some results from the theory of harmonic functions. The basis of these theorems is our ability to solve the first boundary value problem, i.e., our ability to construct a harmonic function with preassigncd boundary values in a region G. In this section we shall apply the results of the first chapter to establish the following special case of this general theorem.


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  1. 1.
    We shall make frequent use of this notation, which was introduced by E. Lindelöf.Google Scholar
  2. 1.
    For the proof of the uniformity one has to first isolate the points of discontinuity of u(ζ) by small arcs that for a given subregion of the interior of G have harmonic measure < ε; the remainder of the sum then differs from the integral by an amount that is smaller than the variation of u (ζ) on the corresponding arcs S n.Google Scholar
  3. 1.
    That means continuous and with a tangent which varies continuously.Google Scholar
  4. 1.
    To achieve this it is sufficient for the boundary function u(ζ) to be continuously differentiate with respect to the are length of Г, if Γ is analytic.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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