Dependence of the boundary-limit condition on the quasiharmonicity constant

Determine whether condition (rs-lim)—the existence of the limit \(\lim_{G\ni y\to x_0}u(y)\) for every bounded \(Q\)-quasiharmonic function in \(G\)—is independent of the quasiharmonicity constant \(Q\).

Background

The paper characterizes removability of compact sets of positive Sobolev $p$-capacity in terms of several equivalent conditions, including a boundary-limit condition for bounded $Q$-quasiharmonic functions. The authors establish implications relating sequential annular quasiconvexity, existence of boundary limits, and local connectedness, but do not determine whether the boundary-limit condition itself depends on the value of $Q$.

Resolving this question would clarify whether that analytic condition is intrinsic to the domain and boundary point or instead varies with the quasiharmonicity constant.