# 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

$u(P(x_j))=0$.

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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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