A counterexample in $L^p$ approximation by harmonic functions

http://dx.doi.org/10.4153/CJM-1997-026-3
Canad. J. Math. 49(1997), 568-582

  Published:1997-06-01
 Printed: Jun 1997

For ${n \over {n-2}}\leq p<\infty$ we show that the conditions $C_{2,q}(G\setminus \dox)=C_{2,q}(G \setminus X)$ for all open sets $G$, $C_{2,q}$ denoting Bessel capacity, are not sufficient to characterize the compact sets $X$ with the property that each function harmonic on $\dox$ and in $L^p(X)$ is the limit in the $L^p$ norm of a sequence of functions which are harmonic on neighbourhoods of $X$.