Let $\phi_1, \phi_2: \mathbb R^2\to\mathbb R$ be solutions of Laplace equation with Dirichlet boundary: $$ \nabla^2\phi_1 = 0,\quad\quad \phi_1(\mathbf x) = a_1 \quad\mbox{if}\quad\mathbf x\in\Omega_1 \\ \nabla^2\phi_2 = 0,\quad\quad \phi_2(\mathbf x) = a_2 \quad\mbox{if}\quad\mathbf x\in\Omega_2 $$
If you wish you can consider the surfaces as being closed, that is, $\Omega = \partial V$. For simplicity, lets assume there's no intersection between them, that is, $\Omega_1\cap\Omega_2 = \emptyset$. You can assume both $\Omega$ to be well behaved in anyway you wish to make things easier. Now, consider the following Dirichlet problem: $$ \nabla^2\phi = 0,\quad\quad \phi(\mathbf x) = \begin{cases} a_1, & \mbox{if} & \mathbf x\in\Omega_1 \\ a_2, & \mbox{if} & \mathbf x\in\Omega_2 \\ \end{cases} $$
I was wondering, is there any kind of relationship between $\phi_1, \phi_2$ and $\phi$? I was interested in $\mathbb R^2$ or $\mathbb R^3$ because it is probably easier. But, if you wish to give an answer about $\mathbb R^n$ go for it. In other words, is there a way to build $\phi$ from $\phi_1$ and $\phi_2$? Also, any resource/book/pdf about this will be welcomed.