I think $\subset$ mean a subset of set, i.e. $D \subset \mathbb R^2$ . and $\in$ mean an element of a set, i.e. if $a$ is an element in $D$, we write $a\in D$.
But in the following $D$ is a domain in $\mathbb R^2$, writing $D\in \mathbb R^2$. Isn't it wrong? Is it not more correct with $D\subset \mathbb R^2$?
Denoted $D\in \mathbb R^2$ for a two-dimensional domain bounded by the closed smooth contour $\Gamma$. $u$ is definied on the domain $D$ and satisfies Laplace's equation: $$ \nabla^2 u(x_1,x_2)=0, \quad \text{in }D$$
To supplement Arnaud Mortier’s comment:
$\Bbb{R}^2$ is a set of ordered pairs. Therefore, if you say $D\in\Bbb{R}^2$, then you are implying that $D$ is itself an ordered pair.
By contrast, the subset relation $\subset$ compares, naturally, two sets. If you say $D\subset\Bbb{R}^2$, then you are implying that $D$ is a set. (Assuming $D\neq\varnothing$, it will contain at least one ordered pair.)
The explanation in the block quote is definitely wrong in this regard.