American Institute of Mathematical Sciences

Advanced
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
[1]

Mihai Mihăilescu, Julio D. Rossi. Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4363-4371. doi: 10.3934/cpaa.2020198
[2]

Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845
[3]

Robert Brooks and Eran Makover. The first eigenvalue of a Riemann surface. Electronic Research Announcements, 1999, 5: 76-81.

[4]

Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089
[5]

Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675
[6]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2589-2618. doi: 10.3934/dcds.2017111
[7]

Zhen Wang, Wei Wu. Bounds for the greatest eigenvalue of positive tensors. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1031-1039. doi: 10.3934/jimo.2014.10.1031
[8]

Leszek Gasiński, Nikolaos S. Papageorgiou. Multiple solutions for a class of nonlinear Neumann eigenvalue problems. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1491-1512. doi: 10.3934/cpaa.2014.13.1491
[9]

Mihai Mihăilescu. An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Communications on Pure & Applied Analysis, 2011, 10 (2) : 701-708. doi: 10.3934/cpaa.2011.10.701
[10]

Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799
[11]

Gang Meng. The optimal upper bound for the first eigenvalue of the fourth order equation. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6369-6382. doi: 10.3934/dcds.2017276
[12]

Wolfgang Arendt, Rafe Mazzeo. Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2201-2212. doi: 10.3934/cpaa.2012.11.2201
[13]

Fang Li, Jerome Coville, Xuefeng Wang. On eigenvalue problems arising from nonlocal diffusion models. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 879-903. doi: 10.3934/dcds.2017036
[14]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[15]

Giacomo Bocerani, Dimitri Mugnai. A fractional eigenvalue problem in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 619-629. doi: 10.3934/dcdss.2016016
[16]

David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems & Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004
[17]

Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008
[18]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[19]

Jean-Michel Rakotoson. Generalized eigenvalue problem for totally discontinuous operators. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 343-373. doi: 10.3934/dcds.2010.28.343
[20]

Gabriele Bonanno, Pasquale Candito, Roberto Livrea, Nikolaos S. Papageorgiou. Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1169-1188. doi: 10.3934/cpaa.2017057

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (151)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences