Recall that a set $E \subset \mathbb{R}^n$ is called (1,2) stable if it satisfies the following: $$ W_0^{1,2}(E) = W_0^{1,2}(E^0), $$ where $E^0$ is the interior of $E$ and $W_0^{1,2}(E)$ is defined by $$ W_0^{1,2}(E) = \{ u \in W_0^{1,2}(\mathbb{R}^n) : u = 0 \text{ on } \mathbb{R}^n \setminus E \}. $$
For equivalent definitions, see a book "Function Spaces and Potential Theory" by Adams and Hedberg (Theorem 11.4.1, for example).
The notion of stability is closely related to the stability of the Dirichlet problem, and to the problem of uniform approximation by harmonic functions.
My question is as follows: is there a relation between Hausdorff dimension and (1,2) stability? For example, is it true that if $\partial E$ has Lebesgue measure 0, then $E$ is (1,2) stable?
Or more generally, can you give some references on the relation between the geometry of a given space and it's (1,2) stability? I am interested in the following question as well: is it true that a Jordan domain is (1,2) stable?
