Status: On

## Mark

considering build up a package for my recent research. Will be back to this post later.

Considering build theory on DtN map for nonhomogeneous source term, it has been thought to have the same property as homogeneous case.

Considering compressive detecting method for point source problem.

Considering particle-based transport equation with inverse source problem.

Categories: Numerics

## Jam

Categories: Accessary, Austin Pavilion

## Spams rampage

Categories: Accessary

## A trip to San Antonio

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Categories: Accessary

## 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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## 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.

## Existence of random zeros::Problems[FINISHED]

February 24, 2013 1 comment

[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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