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C 1 approximation of functions by solutions of second-order elliptic systems on compact sets in ℝ2

Proceedings of the Steklov Institute of Mathematics volume 298, pages 35–50 (2017)Cite this article

Abstract

We consider the problems of C 1 approximation of functions by polynomial solutions and by solutions with localized singularities of homogeneous elliptic second-order systems of partial differential equations on compact subsets of the plane ℝ2. We obtain a criterion of C 1-weak polynomial approximation which is analogous to Mergelyan’s criterion of uniform approximability of functions by polynomials in the complex variable. We also discuss the problem of uniform approximation of functions by solutions of the above-mentioned systems. Moreover, we consider the Dirichlet problem for systems that are not strongly elliptic and prove a result on the lack of solvability of such problems for any continuous boundary data in domains whose boundaries contain analytic arcs.

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Authors and Affiliations

  1. Bauman Moscow State Technical University, Vtoraya Baumanskaya ul. 5/1, Moscow, 105005, Russia

    A. O. Bagapsh & K. Yu. Fedorovskiy

  2. Dorodnicyn Computing Centre, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

    A. O. Bagapsh

  3. Mathematics and Mechanics Faculty, St. Petersburg State University, Universitetskii pr. 28, Peterhof, St. Petersburg, 198504, Russia

    K. Yu. Fedorovskiy

Authors
  1. A. O. Bagapsh
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  2. K. Yu. Fedorovskiy
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Correspondence to A. O. Bagapsh.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 298, pp. 42–57.

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Bagapsh, A.O., Fedorovskiy, K.Y. C 1 approximation of functions by solutions of second-order elliptic systems on compact sets in ℝ2 . Proc. Steklov Inst. Math. 298, 35–50 (2017). https://doi.org/10.1134/S0081543817060037

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  • DOI: https://doi.org/10.1134/S0081543817060037

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