Abstract
Following Escobar (J Funct Anal 150(2):544–556, 1997) and Jammes (Ann l’Inst Fourier 65(3):1381–1385, 2015), we introduce two types of isoperimetric constants and give lower bound estimates for the first nontrivial eigenvalues of Dirichlet-to-Neumann operators on finite graphs with boundary respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bandle, C.: Isoperimetric Inequalities and Applications, Volume 7 of Monographs and Studies in Mathematics. Pitman Advanced Publishing Program, Boston (1980)
Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Universitext. Springer, New York (2012)
Calderón, A.P.: On an inverse boundary value problem. In: Meyer, W.H., Raupp, M.A. (eds.) Seminar on Numerical Analysis and its Applications to Continuum Physics: Rio de Janeiro, pp. 65–73. Sociedade Brasileira de Matematica, Rio de Janeiro (1980)
Chang, K.C.: Spectrum of the 1-Laplacian and Cheeger’s constant on graphs. J. Graph Theory 81(2), 167–207 (2016)
Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Papers Dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton University Press, Princeton (1970)
Chung, F.R.K.: Spectral Graph Theory, Volume 92 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington; by the American Mathematical Society, Providence (1997)
Chang, K.C., Shao, Sihong, Zhang, Dong: The 1-Laplacian Cheeger cut: theory and algorithms. J. Comput. Math. 33(5), 443–467 (2015)
Escobar, J.: The geometry of the first non-zero Stekloff eigenvalue. J. Funct. Anal. 150(2), 544–556 (1997)
Grigor’yan, A.: Analysis on Graphs. Lecture Notes, University of Bielefeld (2009). https://www.math.uni-bielefeld.de/~grigor/aglect.pdf
Jammes, P.: Une inégalité de Cheeger pour le spectre de Steklov. Ann. l’Inst. Fourier 65(3), 1381–1385 (2015)
Kuznetsov, N., Kulczycki, T., Kwasnicki, M., Nazarov, A., Siudeja, B., Poborchi, S., Polterovich, I.: The legacy of Vladimir Andreevich Steklov. Not. Am. Math. Soc. 61(1), 9–23 (2014)
Lawler, G.F.: Random Walk and the Heat Equation. Student Mathematical Library, vol. 55. American Mathematical Society, Providence (2010)
Li, P.: Geometric Analysis. Cambridge Studies in Advanced Mathematics, vol. 134. Cambridge University Press, Cambridge (2012)
Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics, vol. 123. Cambridge University Press, Cambridge (2010)
Taylor, M.E.: Partial Differential Equations. II. Qualitative Studies of Linear Equations, Volume 116 of Applied Mathematical Sciences. Springer, New York (1996)
Uhlmann, G.: Inverse problems: seeing the unseen. Bull. Math. Sci. 4(2), 209–279 (2014)
Acknowledgements
We would like to thank the anonymous referee for his/her helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Jost.
Bobo Hua is supported in part by NSFC, No. 11401106. Zuoqin Wang is supported in part by NSFC, Nos. 11571131 and 11526212.
Rights and permissions
About this article
Cite this article
Hua, B., Huang, Y. & Wang, Z. First eigenvalue estimates of Dirichlet-to-Neumann operators on graphs. Calc. Var. 56, 178 (2017). https://doi.org/10.1007/s00526-017-1260-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-017-1260-3