Existence of random zeros::solution

[UPDATE] This is the last post for this problem.

A couple of weeks ago, I came up with the solution by using moving frame which was learned in Differential Equation class long time ago. Briefly speaking, it is a good piece of work, in constructing special solution, and a nice try with coordinate transformation.

Inverse Problem: on point source.

Now I simply list my trials in probing the path to the solution.

• As I mentioned in last post, I would like to continue using the same technique, with a CGO-like approach, however, after a long time of trials, I gave up with this method. This method will give out a very impressive and concise representation of the problem:

$u\Delta \phi + 2\nabla u \cdot \nabla \phi +k^2 u(n-1) \phi = 0$,

after we multiply $\overline{u}$ to the equation, it will morph into divergence form.

$\nabla(|u|^2\nabla\phi) + k^2|u|^2\phi = 0$.

To my knowledge of this degenerate elliptic equation, we need to apply weighted Sobolev space theory, but unfortunately the theory cannot rule out the existence of singularities, and I came up with a counterexample for that. However, I still believe this method can be promising in 2D case.

• For series form, if we require the coefficients of  expansion as analytic series, i.e.

$\rho = \sum\rho_j \overline{z}^j$, where $\rho_j = \sum c_{kj} z^k$

This can give a recursion formula for the problem, but the analytic property will force the problem to be unsolvable. What a pity.

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