Sectors with Density in Granada

My undergraduate research Geometry Group and I have been having a great summer here in Granada Spain. We’ve been considering planar sectors of angle 0<\theta<\infty with density r^p (p>0) and the isoperimetric problem: to enclose given weighted area with least weighted perimeter. We’ve proved that there are angles 0<\theta_1<\theta_2 \leq\pi such that the minimizer is:

1. for 0<\theta < \theta_1, a circular arc about the origin;

2. for \theta_1<\theta < \theta_2, an unduloid (half-period of a periodic curve normal to both edges of the sector);

3. for \theta_2<\theta, a semicircle through the origin.

We have lots of evidence that \theta_1=\pi/\sqrt{p+1} and \theta_2=\pi(p+2)/(2p+2), but we have not been able to prove it. Can you help us? Check out our arXiv post.

2 Comments

  1. Sean Howe:

    Right now we know that for density r^p the circular arc minimizes up until the π/(p+1) sector. We could improve that to the π/(p/2+1) sector if we could show that in the plane with regular area and perimeter density r^q for 0<q<1 circles about the origin were minimizers. Indeed, the entire conjecture has an equivalent statement in terms of the plane with different area and perimeter densities, but this is a particularly nice case. Furthermore the line p/2+1 is the tangent line at p=0 to √(p+1) so this is the best possible linear (in the denominator) estimate we could get.

  2. Frank Morgan » Blog Archive » Manifolds with Density: Fuller References:

    […] [2012] Alexander Díaz, Nate Harman, Sean Howe, David Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589–619; arXiv.org (2010); see blog posts 1 and 2. […]

Leave a Reply

Your email address will not be published. Required fields are marked *