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admitted function Ahlfors application arbitrary Assertion bordering on end boundary condition Boundary Value Problem Chapter closed subdomain common point component conformal mapping conjugate harmonic function consider constant width h convergence cross cut denote disk distortion theorem domain G elements end CD end values ends,and exists extension principle flow domain flow function flow potential flux fulfilled function u(P Green function Green's formula harmonic function harmonic in G harmonic measure Hayman's heeft immediate consequence interior of G lemma level curves maximum principle means Mixed Boundary Value obtain orthogonality condition parallel piece-wise smooth polygonal pipe proof prove rectangle remark resistance from end Schwarz segments separated in G simply connected domain smooth in G solution source distribution sources and sinks strength of current strip width Subharmonic Functions tends to side theorie triangle inequality upper bound variational problem vector space vertices x,y plane zero