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Beurling’s free boundary value problem in conformal geometry. (English) Zbl 1217.30007

A. Beurling’s celebrated extension of the Riemann mapping theorem [Acta Math. 90, 117–130 (1953; Zbl 0051.06004); Sem. analytic functions 1, 248–263 (1958; Zbl 0099.08302)] as a conformal geometric free boundary value problem arose from the search for functions \(f\) analytic in the unit disc with \(0= f(0)< f'(0)\) for which
\[ \lim_{z\to\zeta}\Big(|f'(z)|-\Phi\big(f(z)\big)\Big)= 0, \quad \zeta\in\{z:|z|= 1\}, \] where \(\Phi: \mathbb{C}\to\mathbb{R}\) is a continuous, positive, bounded function.
The authors noticed that the only known proof of this result contained a number of gaps that seemed inherent in his geometric and approximative approach. In this interesting paper they provide a complete proof of the theorem by combining Beurling’s geometric method (that uses an ingenious geometric version of Perron’s method of subharmonic functions) with a number of new analytic tools, notably \(H^p\)-space techniques from the theory of Riemann-Hilbert-Poincaré problems.
This approach leads to an extension of the theorem to analytic maps with prescribed branching. Also, it allows a complete description of the boundary regularity of solutions in the (generalized) theorem, extending earlier results obtained by B. Gustafsson and H. Shahgholian [J. Reine Angew. Math. 473, 137–179 (1996; Zbl 0846.31005)] by PDE techniques. In addition, they examine the uniqueness question in the extended Beurling-Riemann theorem.
They end with a specific counterexample to Beurling’s original method of proof.

MSC:

30C35 General theory of conformal mappings
30H10 Hardy spaces
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References:

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