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Section
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Mathematics
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Title
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On behavior of solution of boundary value problem for generalized Cauchy-Riemann equation
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Author(-s)
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Il'chukov A.S.a
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Affiliations
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Syktyvkar State Universitya
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Abstract
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The following boundary value problem for generalized Cauchy-Riemann equation in the unit disk $G =\{z \in \mathbb{C}: |z| < 1\}$ is considered in the paper: $\partial_{\overline{z}} w + b(z) \overline{w} = 0,$ $\Re w = g$ on $\partial G,$ $\Im w = h$ at the point $z_0 = 1.$ The coefficient $b(z)$ is chosen from some set $S_P,$ constructed by scales, with $S_P \subsetneq L_2,$ $S_P \not\subset L_q,$ $q > 2.$ The boundary value $g$ is chosen from the space, constructed by a modulus of continuity $\mu$ with some special properties. It is shown that the problem has unique solution $w = w(z)$ in the unit disk $G$ with $w \in C(\overline{G}).$ Moreover, outside the point $z = 0$ the behaviour of the solution $w(z)$ is defined by the same modulus of continuity $\mu;$ it means there is no ''logarithmic effect" for the solution.
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Keywords
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generalized Cauchy-Riemann equation, Dirichlet problem, modulus of continuity
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UDC
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517.53
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MSC
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30E25
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DOI
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10.20537/vm130203
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Received
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1 April 2013
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Language
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Russian
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Citation
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Il'chukov A.S. On behavior of solution of boundary value problem for generalized Cauchy-Riemann equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 2, pp. 27-34.
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References
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