On uniform approximability by solutions of elliptic equations of order higher than two

We consider uniform approximation problems on compact subsets of $\mathbb R^d$, $d>2$, by solutions of homogeneous constant coefficients elliptic equations of order $n>2$. We construct an example showing that in the general case for compact sets with nonempty interior there is no uniform approximability criteria analogous to the well-known Vitushkin's criterion for analytic functions in $\mathbb C$. On the contrary, for nowhere dense compact sets the situation is the same as for analytic and harmonic functions, including instability of the corresponding capacities.