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Find the Green function of the upper half ball $\Omega:=\left\{x\in\mathbb{R}^n|\lVert x\rVert<R, x_n>0\right\}$ (for the Dirichlet boundary value problem of the Laplace equation). Show that the function you found is indeed a Green function. HINT: You are alowed to use the Green function of the ball $$ G_n(x,y)=E_n(x-y)-E_n(\frac{\lVert y\rVert}{R}(x-y*)),~~y*:=\frac{R^2}{\lVert y\rVert^2}y~\text{ for }y\in B_R(0) $$ without having to prove its properties again.

The first step is to find a candidate for the Green function, the second step is to verify its properties. Unfortunately I collapse already in finding a candidate using the given Green function of the ball.

Could you please tell me how I can use the given Green function of the ball to find the Green function of the upper half ball? I do not come along. ;(

Edit

I know that the Green function of the upper half space $O:=\left\{x\in\mathbb{R}^n: x_n>0\right\}$ is given by $$ H_n(x,y)=E_n(x_1-y_1,\ldots,x_{n-1}-y_{n-1},x_n-y_n)-\underbrace{E_n(x_1-y_1,\ldots,x_{n-1}-y_{n-1},x_n+y_n)}_{=:E(x-y')}. $$

Now the upper half ball is the intersection of the the whole ball and $O$.

So my idea for the Green function of the upper half ball is to combine the Green functions of the ball and of $O$ as follows:

$$ F(x,y):=E_n(x-y)-(1_{(\Omega\cap R)\times\Omega}(x,y)E(x-y')+1_{(\overline{\Omega}\setminus R)\times\Omega}E_n(\frac{\lVert y\rVert}{R}(x-y*))), $$ whereat $R:=\left\{x\in\mathbb{R}^n:\lVert x\rVert < R, x_n=0\right\}$.

What do you think?

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    Is there something that you can add to this Green's function that results in the half-ball? Or a way that you can modify this one? – John Dec 20 '13 at 23:48
  • Could you pls have a look on my new idea? –  Dec 22 '13 at 17:35

1 Answers1

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Fix $y\in\Omega$; the function $g_n(x)=G_n(x,y)$ already has the property that $\Delta g_n(x)=\delta(x-y)$, and it also vanishes on the upper hemisphere; the only problem is $g_n\neq 0$ on the plane separating the two half-balls. What's the simplest way you can think of to use $g_n$ to construct a function $h_n$ on $B_R(0)$ that vanishes on this plane? Remember that $h_n$ only needs to satisfy $\Delta h_n(x)=\delta(x-y)$ on the upper half-ball; it can have the "wrong" behaviour on the lower half-ball.

Conversely, given a Green's function $J_n(x,y)$ for $\Omega$ and $j_n(x)=J_n(x,y)$, is there a reasonable way to extend $j_n$ to all of $B_R(0)$? How would this function compare to $g_n$?

Skatche
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  • I have edited my question and added an idea. Would you pls have a look on it? –  Dec 22 '13 at 17:34
  • Sorry for the late reply. Your edit puts you on the right track, but you're still overthinking it. How is the Green's function for the upper half-space related to the Green's function for the entire space? Can you use the same trick to define the Green's function for the half-ball in terms of the Green's function for the ball? – Skatche Jan 16 '14 at 22:42