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?

  • $\begingroup$ You should use the green's function for a ball e.g. math.stackexchange.com/q/1464667/80734 $\endgroup$ – Calvin Khor Apr 22 at 9:29
  • $\begingroup$ 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. $\endgroup$ – mathlettuce Apr 22 at 9:39
  • 1
    $\begingroup$ You are asked to find a radial solution. So express $\Delta$ in spherical coordinates, and you will find an ODE. $\endgroup$ – Calvin Khor Apr 22 at 9:43

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 $B(0, 1)$


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