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

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.