From the class A of analytic functions we proceed to the class H of harmonic functions. The latter are in a sense more flexible than the former and thus easier to treat. In particular the solvability of the Dirichlet problem makes it possible to obtain detailed information on the causes of degeneracy. On the other hand the lack of rigidity results in a great diversity of degeneracy phenomena. To subject them to a systematic treatment it is convenient to start with the class HD of harmonic functions with finite Dirichlet integrals and the corresponding null class 0 HD The close connection with Dirichlet’s principle makes the class 0 HD the most significant one among degeneracy classes related to H.


Riemann Surface Maximum Principle Quasiconformal Mapping Ideal Boundary Finite Genus 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  • L. Sario
    • 1
  • M. Nakai
    • 2
  1. 1.Unversity of CaliforniaCaliforniaUSA
  2. 2.Nagoya UnversityGermany

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