On the boundary values of Riemann's mapping function
Author:
R. J. V. Jackson
Journal:
Trans. Amer. Math. Soc. 259 (1980), 281297
MSC:
Primary 30C20; Secondary 30C40
MathSciNet review:
561837
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Abstract: Classically, the calculus of variations is required to prove the existence of a biholomorphism from the unit disk to a given simplyconnected, smooth domain in the complex plane. Here, the problem is reduced to the solution of an ordinary differential equation along the boundary of the domain. The sole coefficient in this equation is identified with the bounded term in the asymptotic expansion of the Bergman kernel function. It is shown that this coefficient can not depend upon any differential expression involving only the curvature function of the boundary.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005618373
PII:
S 00029947(1980)05618373
Keywords:
Riemann's mapping function,
Plateau's problem,
ordinary differential equations,
potential theory,
Hilbert's transform,
asymptotics of Bergman's kernel function
Article copyright:
© Copyright 1980 American Mathematical Society
