Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem $$ \begin{cases} -\Delta u +\lambda u= 0 & x \in \Omega \\ u = 1 & x \in \partial \Omega \end{cases} $$ with $\lambda >0$. 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)? $$ \begin{cases} -\Delta v + \lambda v = \Delta \mathbf{1}_{\partial \Omega} -\lambda\mathbf{1}_{\partial \Omega} & x \in \Omega \\ v = 0 & x \in \mathbb R^n \setminus \Omega \end{cases} $$ How can this change of variable be made rigorous in the context of viscosity solutions?
This question is motivated by a related post on MathOverflow.
You can trivially make this transformation on $\Omega$. Setting $v=u-1$ you obtain
$$-\Delta v + \lambda v = 0 \ \ \text{in } \Omega,$$
with $v=0$ on $\partial \Omega$. At this point you can extend $v$ to $\mathbb{R}^n\setminus \Omega$ to be identically zero, but I'm not sure what gain you would get from this. The extension will be at most Lipschitz continuous, and definitely is not smooth.