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Let $R > 0$. Determine the radial solution of the problem

\begin{align} - \Delta u(x) & = 1 \text{ if $|x| < R$}\\ u(x) & = 0 \text{ if $|x| = R$} \end{align}

We know the fundamental solution of the Laplace equation in $\mathbb{R}^n$ for n>2:

$\Phi(x) = \frac{1}{(n-2) \cdot w_n} \cdot |x|^{2-n}$,

where $w_n$ denotes the surface area of the unit sphere in $\mathbb{R}^n$.

Additionally we know following theorem:

Suppose $f \in C^2_c(\mathbb{R}^n)$ and let $u = \Phi \ast f$. Then $u \in C^2(\mathbb{R}^n)$ and $- \Delta u = f$ in $\mathbb{R}^n$.

My first guess is, that we have to choose $f = \mathbb{1}_{B(0,R)}$. But this function is not even continuous. And just tacking $f = \mathbb{1}_{\mathbb{R}^n}$ seems to make no sense to me.

Can anyone give me a hint how to approach this problem?

mathlettuce
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  • You should use the green's function for a ball e.g. https://math.stackexchange.com/q/1464667/80734 – Calvin Khor Apr 22 '19 at 09:29
  • Thanks for the hint. I saw this solution in Evans book on pdes. In my course, this question is explicitly asked before the introduction to Green's function. So I would like to solve the problem without this specific function. – mathlettuce Apr 22 '19 at 09:39
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    You are asked to find a radial solution. So express $\Delta$ in spherical coordinates, and you will find an ODE. – Calvin Khor Apr 22 '19 at 09:43
  • Hi again :) whats the (Y) in the title for? Some of your other posts have (Y) and (N)s now too – Calvin Khor Nov 25 '19 at 08:00
  • Hi, oh I didn't know this title changes would alert other people. I was just cleaning up my profile. The title changes are just for me to know on which problems I still have to work/think and which are done. Since you wrote me, I assume I should not have done this. I am sorry if I distracted you from more important things. – mathlettuce Nov 25 '19 at 08:08
  • It didn't alert me, but your posts are now (or at least, were) on the front page :) – Calvin Khor Nov 25 '19 at 08:09
  • Oh no! Is there a way to undo this? – mathlettuce Nov 25 '19 at 08:10
  • Errrrr....I don't know. Probably not? I would just leave things as they are I guess, you could try to flag your post and see if a mod wants to do anything? – Calvin Khor Nov 25 '19 at 08:11
  • Okay, thanks. Do you know a way to change the titles of question so that just I can see my changes? – mathlettuce Nov 25 '19 at 08:13
  • I don't know, sorry. But I have two suggestions: (1) keep an offline list, pen and paper or computer ? (2) Use your profile, which you can freely edit (though you should check how often you can change your profile, if you want to edit it often). Unfortunately your profile is publicly visible – Calvin Khor Nov 25 '19 at 08:18

1 Answers1

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Recipe -

  1. Show that if $u(x) = U(r)$, $r=|x|$, then $$ \Delta u=\frac{\partial^{2} u}{\partial r^{2}}+\frac{N-1}{r} \frac{\partial u}{\partial r}= U'' + \frac{N-1}rU'.$$
  2. Now solve $ U'' + \frac{N-1}rU' = -1$ on $r\in[0,R]$ subject to $ U(R)=0$ and (since $u$ is radial and differentiable at $0$) $U'(0)=0$.
  3. Conclude.

A similar problem - Poisson equation inside a ball $B(0, 1)$

Calvin Khor
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