Archive of Issues
Russia Syktyvkar
Section  Mathematics 
Title  On behavior of solution of boundary value problem for generalized CauchyRiemann equation 
Author(s)  Il'chukov A.S.^{a} 
Affiliations  Syktyvkar State University^{a} 
Abstract  The following boundary value problem for generalized CauchyRiemann 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. 
Keywords  generalized CauchyRiemann equation, Dirichlet problem, modulus of continuity 
UDC  517.53 
MSC  30E25 
DOI  10.20537/vm130203 
Received  1 April 2013 
Language  Russian 
Citation  Il'chukov A.S. On behavior of solution of boundary value problem for generalized CauchyRiemann equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 2, pp. 2734. 
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