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On the existence of the Green’s function for the most general simply connected plane region

1900; American Mathematical Society; Volume: 1; Issue: 3 Linguagem: Inglês

10.1090/s0002-9947-1900-1500539-2

1088-6850

William F. Osgood,

Point processes and geometric inequalities

The problem of mapping the interior of a simply connected plane region T conformally on the interior of a circle depends for its solution on the proof of the existence of the Green's function corresponding to T, i. e., of a function u satisfying the following conditions :(1) Except at a single arbitrarily chosen interior point O of T, u is single valued and harmonic throughout T.By harmonic is meant that u satisfies Laplace's equation, At« = 0 ;(2) At the point O, it becomes discontinuous like log IV , or u = log 1/r + u , where u' is harmonic throughout the neighborhood of O, r denoting the distance of the variable point (x, y) from O ;(3) The boundary values of u are all 0, i. e., when the point (x, y) approaches from the interior of T an arbitrary point of the boundary of T, u approaches the value 0.Hitherto the existence of the Green's function has been established («) for regions T bounded by a finite number of pieces of analytic curves (Schwarz), and (b) for regions T bounded by a finite number of pieces of regular curves (Paraf and Painlevé).By a regular curve is meant one that has a tangent that turns continuously as a variable point traces out the curve.Cf. Picard, Traité, vol.II, chs.Ill, IV, X.In these cases it is shown furthermore that the boundary of T is transformed continuously into the boundary of the circle.In the present paper it is proposed to establish the existence of the Green's function for the most general!simply connected plane region T.Such regions include (a) regions whose boundary is any Jordan curve ; J (/3) regions whose boundary is more * Presented to the .Society June 29, 1900.