[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:

[Reduced case]

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

.

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For Problem 5, I have a solution, but cannot be applied to other ones. Will be recorded next time.

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