Existence of random zeros::Problems[FINISHED]

[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 \mathbb{R}^3, say x_j, j =1 ,\cdots m.  Is there a solution of Helmholtz equation

\Delta u + k^2 u = 0

such that u(x_j) = 0.

  • Problem 2:

What if in \mathbb{R}^d?

  • Problem 3:

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

\Delta u + k^2 n(x) u = 0

where n(x)-1 is supported on a compact domain.

  • Problem 4:

[\mathbb{R}^d case] of the above one.

  • Problem 5:

[Reduced case]

If we cannot find the solution for random zeros,  we define a projection operator P:\mathbb{R}^3\rightarrow \mathbb{R}^2, 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.



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