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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Reflection principle for systems of first order elliptic equations with analytic coefficients

Author: Chung Ling Yu
Journal: Trans. Amer. Math. Soc. 164 (1972), 489-501
MSC: Primary 30A92
MathSciNet review: 0293110
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Abstract: Let T be a simply connected domain of the $ z = x + iy$ plane, whose boundary contains a portion $ \sigma $ of the x-axis. Also let $ A(z,\zeta ),B(z,\zeta ),F(z,\zeta ),\alpha (z),\beta (z)$ and $ \rho (z)$ be holomorphic functions for $ z,\zeta \in T \cup \sigma \cup \bar T$, with $ \alpha (z) - i\beta (z) \ne 0$ for $ z \in \bar T \cup \sigma ,\alpha (z) + i\beta (z) \ne 0$ for $ z \in T \cup \sigma $. Furthermore, we assume that $ \alpha (x)$ and $ \beta (x)$ are real valued functions for $ x \in \sigma $. Our reflection principle states that for any solution $ w = u + iv$ of an equation of the type $ \partial w/\partial \bar z = A(z,\bar z)w + B(z,\bar z)\bar w + F(z,\bar z)$ in T under the boundary condition $ \alpha (x)u + \beta (x)v = \rho (x)$ on $ \sigma ,w$ can be continued analytically across the x-axis, onto the entire mirror image $ \bar T$.

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Keywords: Reflection principle, first order elliptic equation, pseudo-analytic functions, Cauchy-Riemann equations
Article copyright: © Copyright 1972 American Mathematical Society

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