User yimin - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:30:42Z http://mathoverflow.net/feeds/user/23346 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/122191/existence-of-a-function Existence of a function Yimin 2013-02-18T16:25:13Z 2013-04-28T18:22:00Z <p>[also asked here http://math.stackexchange.com/questions/307197]</p> <p>All arguments are in $\mathbb{R}^3$. </p> <p>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 $\Omega$.</p> <p>Assume there are some points $x_j\in\Omega$, where $j=1,2,\cdots.m$.</p> <p>Consider Helmholtz equation</p> <p>$\Delta u + k^2 n(x) u = 0$</p> <p>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 $x_j$]</p> <p>Here you may try to give a method to construct $u$, or prove the existence.</p> <p>And you may take $m=1$ here at first.</p> <p>Thanks.</p> http://mathoverflow.net/questions/122191/existence-of-a-function/124795#124795 Answer by Yimin for Existence of a function Yimin 2013-03-17T16:56:36Z 2013-03-17T16:56:36Z <p>I proved the existence <a href="http://www.ma.utexas.edu/users/yzhong/seminar/IP_Point_Source.pdf" rel="nofollow">here</a>. </p> http://mathoverflow.net/questions/87565/poincare-inequality-for-the-annulus/95929#95929 Answer by Yimin for Poincare inequality for the annulus Yimin 2012-05-03T22:39:24Z 2012-05-03T22:39:24Z <p>If you apply Poincare Inequality on the ball $B\supset A$, thus</p> <p>$|\vert f-\frac{1}{|B|}\int_A f|\vert_{L^2(B)}\le C |\vert\nabla f|\vert_{L^{2}(B)}$</p> <p>and as we know $f=0$ outside $A$, thus</p> <p>$||\frac{1}{|B|}\int f||\le\frac{|A|}{|B|}||f||=\alpha ||f||$</p> <p>By triangular inequality, </p> <p>$|\vert f-\frac{1}{|B|}\int f|\vert_{L^2}\ge (1-\alpha)||f||_{L^2}$.</p> http://mathoverflow.net/questions/122191/existence-of-a-function Comment by Yimin Yimin 2013-02-18T16:28:32Z 2013-02-18T16:28:32Z ok, I will put the link here.