# Dirichlet problem

1. Feb 22, 2010

### Kalidor

1. The problem statement, all variables and given/known data

Let $$B_R = \{ x \in \mathbb{R}^n: |x| < R \}.$$ Calculate the solution of the following Dirichlet problem:

$$-\Delta v = 1$$ in $$B_R$$
$$u = 0$$ on $$\partial B_R$$

Calculate the solution of the problem.

2. Relevant equations

3. The attempt at a solution

I know that the solution must be radial for trivial considerations on the invariance of laplacian under orthogonal transformations and the uniqueness of the solution.
I thought about integrating a Green function for the problem, but what Green function?
There must be an easier way I'm missing.

2. Feb 22, 2010

### Dick

There's a lot easier way. You should be able to guess a solution. What's the laplacian of |x|^2?

3. Feb 22, 2010

### Kalidor

Please don't tell me that the solution is simply $$- \frac{1}{2n}|x|^2 - R^2$$ 'cause I'm gonna shoot myself in the head.

4. Feb 22, 2010

### Dick

You don't have to shoot yourself in the head. It's not right. But it's ALMOST right. Adjust your constant.

5. Feb 22, 2010

### Kalidor

Of course. You don't know how many lines I've written and how much effort I've put into trying to find a Green function to integrate.
Maybe you can tell me if I'm missing something similarly trivial here