Abstract
Let us consider two spaces of harmonic functions. First, the space H(U) of functions harmonic on a bounded open subset U of Rn and continuous to the boundary.
Second, the space H0(K) of functions on a compact subset K of Rn which can be harmonically extended on some open neighbourhood of K. A bounded open subset U of Rn is called stable if the space H(U) is equal to the uniform closure of H0(cl(U)).
In the paper we discussed whether the stability of U is a necessary condition for the equality of systems of functions which are pointwise limits of the spaces H(U) and H0(cl(U)).