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 Viscosity solutions of $(-\Delta)^s u = 0$ in $\Omega $ with non-homogeneous data $u = 1$ in $\mathbb R^n \setminus \Omega$

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?

Because of the answer below, let us consider the same question for $$ \begin{cases} -\Delta u +\lambda u= 0 & x \in \Omega \\ u = 1 & x \in \partial \Omega \end{cases} $$ with $\lambda >0$.

This question is motivated by Viscosity solutions of $(-\Delta)^s u = 0$ in $\Omega $ with non-homogeneous data $u = 1$ in $\mathbb R^n \setminus \Omega$