# Riding on gazelle

God Ever Geometrize
`Status: On`

## Mark

April 22, 2013 Comments off

considering build up a package for my recent research. Will be back to this post later.
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Considering build theory on DtN map for nonhomogeneous source term, it has been thought to have the same property as homogeneous case.
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Considering compressive detecting method for point source problem.
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Considering particle-based transport equation with inverse source problem.

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Categories: Numerics

## Jam

April 13, 2013 Comments off

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Categories: Accessary, Austin Pavilion

## Spams rampage

April 10, 2013 Comments off

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Categories: Accessary

## A trip to San Antonio

March 24, 2013 Comments off

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Categories: Accessary

## Existence of random zeros::solution

March 3, 2013 Comments off

[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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Categories: Complex Analysis, Doodle, Partial Differential Equations Tags: point source

## Existence of random zeros::Explore

March 1, 2013 Comments off

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.

###### Related articles

• Existence of random zeros::Problems (zym8903.wordpress.com)
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Categories: Complex Analysis, Doodle, Partial Differential Equations Tags: Helmholtz equation

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

February 24, 2013

[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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