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

  • Thread starter Kalidor
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In summary, the conversation discusses the solution of a Dirichlet problem involving a Laplacian function. The solution must be radial and can be guessed by considering the invariance of Laplacian under orthogonal transformations. The correct solution involves adjusting a constant and is not as difficult as initially thought. The conversation also mentions a different problem that the speaker has been struggling with and seeks clarification.
  • #1
Kalidor
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Homework Statement



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

[tex] -\Delta v = 1 [/tex] in [tex] B_R [/tex]
[tex] u = 0 [/tex] on [tex] \partial B_R [/tex]

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.
 
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  • #2
There's a lot easier way. You should be able to guess a solution. What's the laplacian of |x|^2?
 
  • #3
Please don't tell me that the solution is simply [tex] - \frac{1}{2n}|x|^2 - R^2 [/tex] 'cause I'm going to shoot myself in the head.
 
  • #4
You don't have to shoot yourself in the head. It's not right. But it's ALMOST right. Adjust your constant.
 
  • #5
Dick said:
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 [Broken]

and finally pull the trigger.
 
Last edited by a moderator:
  • #6
Kalidor said:
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 [Broken]

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.
 
Last edited by a moderator:

1. What is the Dirichlet problem?

The Dirichlet problem is a mathematical problem that involves finding a solution to a partial differential equation (PDE) with boundary conditions specified on the boundary of a given region.

2. What is the equation used to solve the Dirichlet problem?

The equation used to solve the Dirichlet problem is called the Laplace equation, which is given by -\Delta v = 0, where \Delta is the Laplace operator and v is the unknown function.

3. What does the equation -\Delta v = 1 in B_R represent?

The equation -\Delta v = 1 in B_R represents a specific type of Dirichlet problem, where the differential equation is the Laplace equation and the boundary conditions specify that the solution v is equal to 1 inside the region B_R.

4. What does the boundary condition u = 0 on \partial B_R mean?

The boundary condition u = 0 on \partial B_R means that the solution u is equal to 0 on the boundary of the region B_R. This is a type of boundary condition known as a Dirichlet boundary condition.

5. How is the Dirichlet problem solved?

The Dirichlet problem is solved using various techniques, such as separation of variables, Green's functions, and the method of images. These techniques involve finding a solution that satisfies both the differential equation and the specified boundary conditions.

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