[UPDATE:The problem has been solved completely. I posted the rough proof at minfun.info]
Recently I was thinking about the zeros of Helmholtz equation.
- Problem 1:
Suppose we have a bunch of points in , say , . Is there a solution of Helmholtz equation
such that .
- Problem 2:
What if in ?
- Problem 3:
[3D case] What if the media is in-homogeneous, the equation turns out to be
where is supported on a compact domain.
- Problem 4:
[ case] of the above one.
- Problem 5:
If we cannot find the solution for random zeros, we define a projection operator , maps points onto a plane.
Then can we find a solution to the Helmholtz equation such that
For Problem 5, I have a solution, but cannot be applied to other ones. Will be recorded next time.