# American Institute of Mathematical Sciences

January  2007, 17(1): 1-19. doi: 10.3934/dcds.2007.17.1

## On the first positive Neumann eigenvalue

 1 School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 2 Mathematics Department, Tulane University, New Orleans, LA 70118, United States

Received  September 2006 Published  October 2006

We study the first positive Neumann eigenvalue $\mu_1$ of the Laplace operator on a planar domain $\Omega$. We are particularly interested in how the size of $\mu_1$ depends on the size and geometry of $\Omega$. A notion of the intrinsic diameter of $\Omega$ is proposed and various examples are provided to illustrate the effect of the intrinsic diameter and its interplay with the geometry of the domain.
Citation: Wei-Ming Ni, Xuefeng Wang. On the first positive Neumann eigenvalue. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 1-19. doi: 10.3934/dcds.2007.17.1
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