# Poisson equation $-\Delta u = 1$ with Dirichlet condition

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 $$\mathbb{R}^n$$.

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?

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

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