Poisson's Formula for the half space

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Homework Statement

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Relevant Equations

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Hi all, thanks in advance for any constructive feedback.

Definition:
If $x\in R^n\backslash \{0\}$ then the map $\Lambda$ takes the point $x$ into $\bar{x}\in R^n\backslash R^+_3$ given by $\bar{x}=\{x_1,x_2,-x_3\}$

We take the reflected point $\bar{x}$ and the fundamental solution
$$\Phi=\frac{1}{4\pi ||x||}$$ then
$$\phi^x(y)=\frac{1}{4\pi ||y-\bar{x}||}$$
and hence $$G(x,y)=\frac{1}{4\pi ||y-x||}-\frac{1}{4\pi ||y-\bar{x}||}$$ is the green's function for $R^+_3$

The outward pointing unit normal vector field on the boundary is given by a function $\nu$ that assigns a unit vector $-\vec{e}_3$ to every point on $x_3=0$.

The differential operator $$\frac{\partial}{\partial \nu}\rightarrow\frac{-\partial }{\partial x_3}$$

The derivative of $G(x,y)$ in the direction of the vector field $\nu$ is given by
$$\frac{\partial }{\partial \nu}G(x,y)= \frac{-\partial }{\partial x_3}\Big[\Phi (y-x)-\Phi(y-\bar{x})\Big]=\frac{-\partial }{\partial x_3}\Big[\frac{1}{4\pi ||y-x||}-\frac{1}{4\pi ||y-\bar{x}||}\Big]$$

Have I made mistakes so far, and can I continue working?

 

[docnet]

update with "progress"

Definition:
If $x\in R^n\backslash \{0\}$, then the map $$\Lambda(x_1,x_2,x_3)=(x_1,x_2,-x_3)$$ takes every point $x$ into $\bar{x}=\{x_1,x_2,-x_3\}\in R^n\backslash R^3_+$

To solve the problem
$$
\begin{cases}
& \Delta u=f \quad \text{in} \quad R^3_+\\
& u= g \quad \text{on} \quad \partial R^3_+
\end{cases}
$$
We consider the reflected point $\bar{x}$ and the fundamental solution $\Phi=\frac{1}{4\pi ||x||}$. The following problem
$$
\begin{cases}
& \Delta \phi^x=0 \quad \text{in} \quad R^3_+\\
& \phi^x(y)= \Phi(y-x) \quad \text{on} \quad \partial R^3_+
\end{cases}
$$
gives the corrector function
$$\phi^x(y)=\frac{1}{4\pi ||y-\bar{x}||}$$
and hence the green's function is $$G(x,y)=\frac{1}{4\pi ||y-x||}-\frac{1}{4\pi ||y-\bar{x}||}$$
The outward-pointing unit normal vector field on the set $\{x_3=0\}$ is given by a function $\nu$ that assigns the unit vector $-\vec{e}_3$ to every point on the set $\{x_3=0\}$. Let us take the differential operator $$\frac{\partial}{\partial \nu}\Rightarrow\frac{\partial }{\partial (-\vec{e}_3)}$$
and compute the derivative of $G(x,y)$ in the direction $\nu$ on the set $\{x_3=0\}$ in the flat Euclidean connection
$$\frac{\partial }{\partial (-\vec{e}_3)}G(x,y)\Rightarrow \frac{1}{4\pi}\frac{\partial }{\partial (-\vec{e}_3)}\Big[\frac{1}{ ||y-x||}-\frac{1}{||y-\bar{x}||}\Big]$$