Beurling’s free boundary value problem in conformal geometry
Tóm tắt
The subject of this paper is Beurling’s celebrated extension of the Riemann mapping theorem [5]. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem (due to Beurling) contains a number of gaps which seem inherent in Beurling’s geometric and approximative approach. We provide a complete proof of the Beurling-Riemann mapping theorem by combining Beurling’s geometric method with a number of new analytic tools, notably H
p
-space techniques and methods from the theory of Riemann-Hilbert-Poincaré problems. One additional advantage of this approach is that it leads to an extension of the Beurling-Riemann mapping theorem for analytic maps with prescribed branching. Moreover, it allows a complete description of the boundary regularity of solutions in the (generalized) Beurling-Riemann mapping theorem extending earlier results that have been obtained by PDE techniques. We finally consider the question of uniqueness in the extended Beurling-Riemann mapping theorem.
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