$C^m$ approximation of functions by solutions of second-order elliptic systems on compact sets in the plane

A. O. Bagapsh ; K. Yu. Fedorovskiy

Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 7-17.

Résumé

This paper is a brief survey of the recent results in problems of approximating functions by solutions of homogeneous elliptic systems of PDEs on compact sets in the plane in the norms of $C^m$ spaces, $m\geq0$. We focus on general second-order systems. For such systems the paper complements the recent survey by M. Mazalov, P. Paramonov, and K. Fedorovskiy (2012), where the problems of $C^m$ approximation of functions by holomorphic, harmonic, and polyanalytic functions as well as by solutions of homogeneous elliptic PDEs with constant complex coefficients were considered.

Détail

Mots-clés : elliptic equation, $s$-dimensional Hausdorff content
Keywords: second-order elliptic system, $C^m$ approximation, $\kappa _{m,\tau ,\sigma }$-capacity, Vitushkin localization operator.

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