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

On the first positive Neumann eigenvalue

Wei-Ming Ni 1, and  Xuefeng Wang 2,

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.
Keywords: first positive eigenvalue, Neumann eigenvalue problem, domain-monotonicity, intrinsic diameter, diffusion..
Mathematics Subject Classification: 35P15, 35K0.
Citation: Wei-Ming Ni, Xuefeng Wang. On the first positive Neumann eigenvalue. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 1-19. doi: 10.3934/dcds.2007.17.1
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