PDE with an inequality constrain

[BlackTulip]

Hi everybody,

For part of my research, I need to solve an elliptic PDE like:

Δu - k * u = 0,

subject to : 0≤ u(x,y) ≤ 1.0

where k is a positive constant.



Can anyone tell me how I can solve this problem?


Thanks in advance for your help.

 

[haruspex]

Do you mean ∇²u - k*u = 0?

 

[BlackTulip]

Yes, It's the same thing: http://en.wikipedia.org/wiki/Laplace's_equation

 

[haruspex]

Well, the usual approach is separation of variables to obtain a general solution. Normally one has completely specified boundary conditions, whereas you only have bounds, so there will be multiple solutions, and its not obvious how to characterise those that satisfy the bounds. Is that where you're stuck?

 

[BlackTulip]

I do have some Dirichlet boundary conditions. My problem is that how to formulate this PDE with some inequality constraints. Do you have a suggestion for that?

Thanks for your attention.

 

[haruspex]

No idea if this is feasible, but how if you were to replace u by a function that could only be in that range, e.g. substitute u = exp(-v²)? Tricky part might be finding such a substitution that still allows you to solve the PDE.