An Asymptotic Faber-Krahn Inequality for the Combinatorial Laplacian on Z^2

Details An Asymptotic Faber-Krahn Inequality for the Combinatorial Laplacian on Z^2 An Asymptotic Faber-Krahn Inequality for the Combinatorial Laplacian on Z^2

2010-08-24

arxiv.org/abs/1008.4092v1

Mathematics

Spectral Theory

64 pages, Undergraduate Honors Thesis

Scientific paper

The Faber-Krahn inequality states that among all open domains with a fixed volume in R^n, the ball minimizes the first Dirichlet eigenvalue of the Laplacian. We study an asymptotic discrete analogue of this for the combinatorial Dirichlet Laplacian acting on induced subgraphs of Z^2. Namely, an induced subgraph G with n vertices is called a minimizing subgraph if it minimizes the first eigenvalue of the combinatorial Dirichlet Laplacian among all induced subgraphs with n vertices. Consider an induced subgraph G and take the interior of the union of closed squares of area 1 about each point of G. Let G* denote this domain scaled down to have area 1. Our main theorem states that if {G_n} is a sequence of minimizing subgraphs where each G_n has n vertices, then after translation the measure of the symmetric difference of G_n* and the unit disk converges to 0.

Affiliated with

Also associated with

No associations

Rating

An Asymptotic Faber-Krahn Inequality for the Combinatorial Laplacian on Z^2 does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with An Asymptotic Faber-Krahn Inequality for the Combinatorial Laplacian on Z^2, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and An Asymptotic Faber-Krahn Inequality for the Combinatorial Laplacian on Z^2 will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-394213