I am curious about the regularity of the solution in $H^1$ of the following PDE: In $\mathbb{R}^n$ let $B = B(0, 1)$, $\Sigma = \{x_n = 0\}$ and $T = B(0, 1/2) \cap \Sigma$. Consider \begin{align} \Delta u &= 0 \text{ in } B_+ \cup B_- \cup T \\ u &= 1 \text{ on } \partial B \setminus \Sigma \\ \frac{\partial u}{\partial n} &= 0 \text{ on both sides of } B \cap (\Sigma \setminus \bar{T}) \end{align} where $B_+$ and $B_+$ are the upper and lower half-balls respectively. In other words, the domain consists of two open half-balls touching along an $(n - 1)$-dimensional ball in their flat surfaces.
The variational formulation is as follows: $$ \int_B \nabla u \nabla v \, dx = 0$$ for all $v \in H^1(B_+ \cup B_- \cup T)$ having trace equal to $0$ on $\partial B \setminus \Sigma$.
By the answer given here, the solution will be smooth everywhere on the boundary except perhaps on the round edges of the half-balls. Can we say that the solution is smooth up to the boundary?