Article
Ufa Mathematical Journal
Volume 4, Number 4, pp. 105-115
On uniform approximability by solutions of elliptic equations of order higher than two
Mazalov M. Ya.
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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.