User yimin - MathOverflow

Existence of a function

2013-02-18T16:25:13Z

[also asked here http://math.stackexchange.com/questions/307197]

All arguments are in $\mathbb{R}^3$.

Suppose $n(x)$ is a smooth function where $\mathbf{supp}(n(x)-1)$ is a compact set $\Omega$. i.e. $n(x) = 1$ when $x$ is outside $\Omega$.

Assume there are some points $x_j\in\Omega$, where $j=1,2,\cdots.m$.

Consider Helmholtz equation

$\Delta u + k^2 n(x) u = 0$

And I want to know if there is a function $u$ satisfies the equation and also vanishes at $x_j$. i.e. $u(x_j) = 0$. [Certainly $u$ can have a manifold of zeros, here I just restrict it on $x_j$]

Here you may try to give a method to construct $u$, or prove the existence.

And you may take $m=1$ here at first.

Thanks.

Answer by Yimin for Existence of a function

2013-03-17T16:56:36Z

I proved the existence here.

Answer by Yimin for Poincare inequality for the annulus

2012-05-03T22:39:24Z

If you apply Poincare Inequality on the ball $B\supset A$, thus

$|\vert f-\frac{1}{|B|}\int_A f|\vert_{L^2(B)}\le C |\vert\nabla f|\vert_{L^{2}(B)}$

and as we know $f=0$ outside $A$, thus

$||\frac{1}{|B|}\int f||\le\frac{|A|}{|B|}||f||=\alpha ||f||$

By triangular inequality,

$|\vert f-\frac{1}{|B|}\int f|\vert_{L^2}\ge (1-\alpha)||f||_{L^2}$.

Comment by Yimin

2013-02-18T16:28:32Z

ok, I will put the link here.