Page 809 here shows Dirichlet's condition:
$\begin{cases} -\nabla u = \rho & \text{when }A\text{ is an inner point} \\ u=0 & \text{when on the border of } \partial A \\ \end{cases}$
and there are certain extremely vague statements such as the top of the page 809 or I just cannot make head-or-tail of the language.
I am trying to translate the page 809 here to English (may contain mistakes)
If vector field $\bar{F}$ with source $\rho$ is known and we know that $F$ is gravitation field, so some potential $u$ exists so that $\bar{F}=-\nabla u$ (so it is with electricity field and gravitation field), so field and source relationship is $\bar{F}=\rho$ that can be written as
$\Delta u=\rho, \tag{1}$
so called Poisson equation. Points where $\bar{F}$ is sourceless, i.e. $\nabla \cdot \bar{F}=0$ are satisfied with the Laplace equation
$\Delta u=0,$
also known as harmonic.
The Laplace and Poisson equations are commonly in boundary value problems, where we consider open set $A$ or as-usually-said area $A$ ($A\subset\mathbb R^2$ or $A\subset\mathbb R^3$) -- and in the boundary some border condition. -- For example, the boundary condition can be so-called (homogeneous) Dirichlet's condition, so we get
$\begin{cases} -\nabla u = \rho & \text{when }A\text{ is an inner point} \\ u=0 & \text{when on the border of } \partial A \\ \end{cases}$
This can be shown with certain rules -- [cannot understand this part]. Usually this kind of situations are solved numerically.
Reflecting
Now, it is extremely hard to see what is the main message here. If I can understand right, there are something called boundary-value problems that and Dirichlet/Laplace equations may be typical examples of them, somehow with physical restrictions. The part $(1)$ is something I cannot understand at all. It means $\Delta u = \rho=\bar{F}=-\nabla u$ so $\Delta u=-\nabla u$. Sorry but is there any sense with $\Delta u=-\nabla u$? I am gasping here sorry, no way -- something is here skewed -- could someone explain? Please, contain references so I could find some peer-reviewed material in English to compare things, perhaps understanding this better.