Solve Dirichlet Problem: -\Delta v = 1 in B_R, u = 0 on \partial B_R

[Kalidor]

Homework Statement

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.

Homework Equations

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.

 

[Dick]

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

 

[Kalidor]

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

 

[Dick]

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

 

[Kalidor]

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

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

https://www.physicsforums.com/showthread.php?t=380559

and finally pull the trigger.

 

[Dick]

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

https://www.physicsforums.com/showthread.php?t=380559

and finally pull the trigger.

Sorry, that one's not so clear to me as the other one. I'll try and give it some thought.