What is $\Omega$ in the context of Poisson's equation? I have recently started a new course on PDE's and have already stumbled on an example that I'm struggling to understand:

The main aspect of this example that I am struggling to wrap my head around is not in the proof, it's in the way that $\Omega$ is used in the second and third lines.
Does $\Omega$ denote the solution space? i.e. Does $\Omega = (x,y,z)$ and $\partial \Omega = (\partial x , \partial y , \partial z)$ (if the problem is to be solved in 3 dimensions)?
Furthermore, if it is the case that $\Omega$ denotes the solution space for the problem, why is $\partial \Omega$ the boundary of the solution space?

For PDEs, it is very common to denote the domain of the PDE with $\Omega$.
Common examples are a circle/ball 
$$
 \Omega = B_1(0) := \{ x\in \mathbb R^n : \|x\| < 1 \}
$$
or a square
$$
 \Omega = (0,1)\times (0,1)
$$
In this case $\partial\Omega$ denotes the boundary of the domain, and not partial derivatives!
For example, in the case of $\Omega=B_1(0)$
we have
$$
 \partial\Omega = \partial B_1(0) := \{ x\in \mathbb R^n : \|x\| = 1 \}
$$
