Criteria for $C^1$-approximability of functions on compact sets in~${\mathbb{R}}^N$, $N\geqslant 3$, by solutions of~second-order homogeneous elliptic equations

P. V. Paramonov

Geodesic

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Criteria for $C^1$-approximability of functions on compact sets in~${\mathbb{R}}^N$, $N\geqslant 3$, by solutions of~second-order homogeneous elliptic equations

P. V. Paramonov

Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 483-505

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Résumé

We obtain capacitive criteria for the approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients in the norm of a Whitney-type $C^1$-space on a compact set in $\mathbb{R}^N$, $N \geqslant 3$. The case $N=2$ was studied in a recent paper by the author and Tolsa. For $C^1$-approximations by harmonic functions (with any $N$), weaker criteria were earlier found by the author. We establish some metric properties of the capacities considered.

Keywords: $C^1$-approximation, second-order elliptic equation, Vitushkin's localization operator, $\mathcal{L}C^1$-capacity, semi-additivity problem.
Mots-clés : $L$-oscillation, $p$-dimensional Hausdorff content

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     title = {Criteria for $C^1$-approximability of functions on compact sets in~${\mathbb{R}}^N$, $N\geqslant 3$, by solutions of~second-order homogeneous elliptic equations},
     journal = {Izvestiya. Mathematics },
     pages = {483--505},
     publisher = {mathdoc},
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     number = {3},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a9/}
}

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P. V. Paramonov. Criteria for $C^1$-approximability of functions on compact sets in~${\mathbb{R}}^N$, $N\geqslant 3$, by solutions of~second-order homogeneous elliptic equations. Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 483-505. http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a9/