[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.

Suppose we have a bunch of points in , say , . Is there a solution of Helmholtz equation

such that .

What if in ?

[3D case] What if the media is in-homogeneous, the equation turns out to be

where is supported on a compact domain.

[ case] of the above one.

[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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## Published by YiMin

Traveller, in math.
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