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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.

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The equation in the reference is not $\bar F=\rho$, it is $\nabla \bar F=\rho$. –  Byron Schmuland Mar 22 '12 at 23:09
    
This en.wikipedia.org/wiki/Laplace_operator may also help. –  Byron Schmuland Mar 22 '12 at 23:18
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If vector field $\bar{F}$ with source $\rho$ is known and we know that $F$ is a gravitational field, so some potential $u$ exists so that $\bar{F}=-\nabla u$ (as it is with electricity fields and gravitation fields), so field and source relationship is $\nabla \cdot \bar{F}=\rho$ that can be written as

$$\Delta u= -\rho, $$

the so called Poisson equation. Points where $\bar{F}$ is sourceless, i.e. $\nabla \cdot \bar{F}=0,$ satisfy the Laplace equation

$$\Delta u=0,$$

in which case $u$ is known as harmonic.

The Laplace and Poisson equations are commonly encountered in boundary value problems, where we consider open set $A$ or as-usually-said domain $A$ ($A\subset\mathbf R^2$ or $A\subset\mathbf R^3$) -- and in the border $\partial A$ some boundary condition holds. -- For example, the boundary condition can be the so-called (homogeneous) Dirichlet's condition, where we have

$$ -\Delta u = \rho \; \; \mbox{when at an interior point} $$ $$ u=0 \; \; \mbox{when on the boundary } \partial A $$

This problem has a unique solution under fairly mild restrictions,such as when $\rho$ is bounded, locally Hölder continuous on $A,$ and every point of $\partial A$ is regular with respect to the Laplacian. A simple sufficient condition for regularity of $A$ is that an exterior sphere condition holds at each point of $\partial A.$

NOTE: I added the last bit out of Elliptic Partial Differential Equations of Second Order by Gilbarg and Trudinger. Vector fields that can be written as the gradient of some function are called "conservative" fields. In $\mathbb R^3,$ the first thing to check is that the curl of a field is zero, in which case it is (locally) the gradient of a function.

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