Existence of random zeros::Explore

Last time I considered the problems related to  ‘existence of random zeros’. And for No.5 problem, we can find a solution.

The governing equation:

\Delta u + k^2 u =0,

and u(P(x_j)) = 0. Here P is a projection operator onto the xy plane.

Our solution is u = \prod_{j=1}^m (x+iy - a_j -ib_j) e^{-ikz} = \Phi e^{-ikz}, here P(x_j) = (a_j,b_j), since \Phi is analytic, then \Delta \Phi = 0 .

Thus I only have to look at \mathbb{R}^3 case, for high dimension spaces, we just project the points onto a lower dimensional one.

For No.4, we just need to consider the solution to No.3

I do not think the randomness can be achieved for No.1 and No.2, but proof needs more work to do.



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