Computational Methods and Function Theory 2 (2002), No. 1, 175--190
Copyright Heldermann Verlag 2002
Gunter Semmler
semmler@math.tu-freiberg.de, Institute of Applied Mathematics I, University of Mining and Technology, 09596 Freiberg, Germany.
Elias Wegert
wegert@math.tu-freiberg.de, Institute of Applied Mathematics I, University of Mining and Technology, 09596 Freiberg, Germany.
Restrictions imposed on the boundary values of holomorphic functions induce restrictions on their values at interior points. The paper is devoted to the following related question: Let $A$ be a subclass of the Hardy space $H^1$ in the complex unit disk $\mathbb{D}$ and for each $t\in \partial\mathbb{D}$ let the complex plane be divided into an upper und a lower domain by some curve $M_t$. If we know that the boundary values of holomorphic functions $w_+=u_+ +\imag\,v_+$ and $w_-=u_- +\imag\,v_-$ in $A$ lie in the upper and the lower domain respectively, what can we conclude about the relative position of $w_+(0)$ and $w_-(0)$? The problem is studied for several classes $A$ and with different assumptions on the separating curves $M_t$. A number of counterexamples illustrates the limitations of the results. One main tool in the investigations is a non-linear boundary value problem of Riemann-Hilbert type, which is also of independent interest. Using an approximation procedure and an argument based on normal families we extend earlier results on existence and uniqueness of solutions for smooth problems to the case of piecewise continuous boundary conditions.
Keywords: Holomorphic functions, normal family, separation principle, Riemann-Hilbert problem.
MSC 2000: 30C99, 30E25, 30C75.
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