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A local result for systems of Riemann-Hilbert barrier problems


Author: Kevin F. Clancey
Journal: Trans. Amer. Math. Soc. 200 (1974), 315-325
MSC: Primary 45E05
MathSciNet review: 0380307
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Abstract: The Riemann-Hilbert barrier problem (for $ n$ pairs of functions)

$\displaystyle G{\Phi ^ + } = {\Phi ^ - } + g$

is investigated for the square integrable functions on a union of analytic Jordan curves $ C$ bounding a domain in the complex plane. In the special case, where at each point $ {t_0}$ of $ C$ the symbol $ G$ has at most two essential cluster values $ {G_1}({t_0}),{G_2}({t_0})$, then the condition $ \det [(1 - \lambda ){G_1}({t_0}) + \lambda {G_2}({t_0})] \ne 0$, for all $ {t_0}$ in $ C$ and all $ \lambda (0 \leqslant \lambda \leqslant 1)$, implies the Riemann-Hilbert operator is Fredholm. In the case, where for some $ {t_0}$ in $ C$ and some $ {\lambda _0}(0 \leqslant {\lambda _0} \leqslant 1),\det [(1 - {\lambda _0}){G_1}({t_0}) + {\lambda _0}{G_2}({t_0})] = 0$, the Riemann-Hilbert operator is not Fredholm. An application is given to systems of singular integral equation on $ {L^2}(E)$, where $ E$ is a measurable subset of $ C$.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0380307-6
Article copyright: © Copyright 1974 American Mathematical Society