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## Archive of Issues

Russia Syktyvkar
Year
2013
Issue
2
Pages
27-34
 Section Mathematics Title On behavior of solution of boundary value problem for generalized Cauchy-Riemann equation Author(-s) Il'chukov A.S.a Affiliations Syktyvkar State Universitya Abstract 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. Keywords generalized Cauchy-Riemann 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 Cauchy-Riemann equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 2, pp. 27-34. References Vekua I.N. Obobshchennye analiticheskie funktsii (Generalized analytic functions), Moscow: Nauka, 1988, 512 p. Mikhailov L.G. A new class of singular integral equations and its applications to differential equations with singular coefficients, Berlin: Akademie-Verlag, 1970. Usmanov Z.D. Generalized Cauchy–Riemann systems with a singular point, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 85, Longman, Harlow, 1997. Tungatarov A. On the theory of the Carleman–Vekua equation with a singular point, Russ. Acad. Sci. Sb. Math., 1994, vol. 78, no. 2, pp. 357–365. Bliev N. Generalized analytic functions in fractional spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 86, Longman, Harlow, 1997. Reissig M., Timofeev A. Dirichlet problems for generalized Cauchy–Riemann systems with singular coefficients, Complex variables, 2005, vol. 73, no. 1–2, pp. 653–672. Timofeev A.Yu. Boundary problem for the generalized Cauchy–Riemann equation in the spaces, described by the modulus of continuity, Ufa Mathematical Journal, 2012, vol. 4, no. 1, pp. 146–152. Ilchukov A.S., Timofeev A.Yu. Dirichlet problem for holomorphic functions in spaces, described by modulus of continuity with predefined conditions, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 1, pp. 58–65. Tutschke W. Vorlesungen uber partielle Differentialgleichungen. Klassische, funktionalanalytische und komplexe Methoden, Leipzig: Teubner-Texte zur Mathematik, 1978. 193 s. Full text