Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem 
$$
\begin{cases}
-\Delta u = 0 & x \in \Omega \\
u = 1 & x \in \partial \Omega
\end{cases}
$$
Does it make sense to use the change of variables $v = u-\mathbf{1}_{\partial \Omega} $ to reduce the problem to the following one (with a source term but homogeneous boundary data)? 
$$
\begin{cases}
-\Delta v = \Delta \mathbf{1}_{\partial \Omega} & x \in \Omega \\
v = 0 & x \in \mathbb R^n \setminus \Omega
\end{cases}
$$
How can this change of variable be made rigorous in the context of viscosity solutions? 

This question is motivated by https://mathoverflow.net/questions/407520/viscosity-solutions-of-deltas-u-0-in-omega-with-non-homogeneous-dat