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 Tr. Mat. Inst. Steklova, 2002, Volume 236, Pages 142–152 (Mi tm284)

On the Boundary Properties of Solutions to the Generalized Cauchy–Riemann Equation

E. P. Dolzhenkoa, V. I. Danchenkob

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Vladimir State University

Abstract: The paper continues the study of boundary properties of polyanalytic functions and their holomorphic components started by the authors in 1998. Integral formulas for polyanalytic functions and their components as well as some generalizations of the Cauchy integral formula to polyanalytic functions are obtained. For polyanalytic and polyharmonic functions, special mean value theorems and a local maximum principle are proved. Some growth estimates for formal derivatives of polyanalytic (in particular, polyrational) functions and for their components near the boundary of their domain are found. For biharmonic functions, necessary conditions for a local extremum are pointed out.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 236, 132–142

Bibliographic databases:
UDC: 517.53
Received in December 2000

Citation: E. P. Dolzhenko, V. I. Danchenko, “On the Boundary Properties of Solutions to the Generalized Cauchy–Riemann Equation”, Differential equations and dynamical systems, Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko, Tr. Mat. Inst. Steklova, 236, Nauka, MAIK «Nauka/Inteperiodika», M., 2002, 142–152; Proc. Steklov Inst. Math., 236 (2002), 132–142

Citation in format AMSBIB
\Bibitem{DolDan02} \by E.~P.~Dolzhenko, V.~I.~Danchenko \paper On the Boundary Properties of Solutions to the Generalized Cauchy--Riemann Equation \inbook Differential equations and dynamical systems \bookinfo Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko \serial Tr. Mat. Inst. Steklova \yr 2002 \vol 236 \pages 142--152 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm284} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1931014} \zmath{https://zbmath.org/?q=an:1079.30056} \transl \jour Proc. Steklov Inst. Math. \yr 2002 \vol 236 \pages 132--142 

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This publication is cited in the following articles:
1. M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskiy, “Conditions for $C^m$-approximability of functions by solutions of elliptic equations”, Russian Math. Surveys, 67:6 (2012), 1023–1068
2. V. I. Danchenko, “Cauchy and Poisson formulas for polyanalytic functions and applications”, Russian Math. (Iz. VUZ), 60:1 (2016), 11–21
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