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On a method of approximation by gradients

Author: M. B. Dubashinskiĭ
Translated by: the author
Original publication: Algebra i Analiz, tom 25 (2013), nomer 1.
Journal: St. Petersburg Math. J. 25 (2014), 1-22
MSC (2010): Primary 41A30, 41A63
Published electronically: November 20, 2013
MathSciNet review: 3113426
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Abstract: The subject of this study is the possibility of approximation of a continuous vector field on a compact set $ K\subset \mathbb{R}^n$ by gradients of smooth functions defined on the entire $ \mathbb{R}^n$. A method is obtained that yields either approximation or an obstruction for it. This method does not involve the Hahn-Banach theorem and is based on solving a quasilinear elliptic equation in partial derivatives. A discrete analog of the above problem is studied, namely, the problem of approximation by gradients on a finite oriented graph. A stepwise algorithm is suggested in this case.

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Additional Information

M. B. Dubashinskiĭ
Affiliation: Chebyshev Laboratory, St. Petersburg State University, 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia

Keywords: Approximation by gradients, solenoidal vector charge, direct methods, analysis on graphs
Received by editor(s): September 30, 2012
Published electronically: November 20, 2013
Additional Notes: The author was supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government, grant 11.G34.31.0026
Article copyright: © Copyright 2013 American Mathematical Society