Mixed eigenvalues of <i>p</i>-Laplacian

doi:10.1007/s11464-015-0375-0

Front. Math. China

2015, Vol. 10

Issue (2) : 249-274 https://doi.org/10.1007/s11464-015-0375-0

RESEARCH ARTICLE

Mixed eigenvalues of p-Laplacian

Mu-Fa CHEN¹,Lingdi WANG^1,^2,^*(

),Yuhui ZHANG¹

1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
2. School of Mathematics and Information Sciences, Henan University, Kaifeng 475004, China

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Abstract

The mixed principal eigenvalue of p-Laplacian (equivalently, the optimal constant of weighted Hardy inequality in L^p space) is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper.

Keywords p-Laplacian Hardy inequality in L^p space mixed boundaries explicit estimates eigenvalue approximating procedure

Corresponding Author(s): Lingdi WANG

Issue Date: 12 February 2015

Cite this article:

Mu-Fa CHEN,Lingdi WANG,Yuhui ZHANG. Mixed eigenvalues of p-Laplacian[J]. Front. Math. China, 2015, 10(2): 249-274.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-015-0375-0
https://academic.hep.com.cn/fmc/EN/Y2015/V10/I2/249

1	Ca?ada A I, Drábek P, Fonda A. Handbook of Differential Equations: Ordinary Differential Equations, Vol 1. North Holland: Elsevier, 2004, 161-357
2	Chen M F. Eigenvalues, Inequalities, and Ergodic Theory. New York: Springer, 2005
3	Chen M F. Speed of stability for birth-death process. Front Math China, 2010, 5(3): 379-516 https://doi.org/10.1007/s11464-010-0068-7
4	Chen M F, Wang L D, Zhang Y H. Mixed principal eigenvalues in dimension one. Front Math China, 2013, 8(2): 317-343 https://doi.org/10.1007/s11464-013-0229-6
5	Chen M F, Wang L D, Zhang Y H. Mixed eigenvalues of discrete p -Laplacian. Front Math China, 2014, 9(6): 1261-1292 https://doi.org/10.1007/s11464-014-0374-6
6	Drábek P, Kufner A. Discreteness and simplicity of the spectrum of a quasilinear Strum-Liouville-type problem on an infinite interval. Proc Amer Math Soc, 2005, 134(1): 235-242 https://doi.org/10.1090/S0002-9939-05-07958-X
7	Jin H Y, Mao Y H. Estimation of the optimal constants in the Lp-Poincaré inequalities on the half line. Acta Math Sinica (Chin Ser), 2012, 55(1): 169-178 (in Chinese)
8	Kufner A, Maligranda L, Persson L. The prehistory of the Hardy inequality. Amer Math Monthly, 2006, 113(8): 715-732 https://doi.org/10.2307/27642033
9	Kufner A, Maligranda L, Persson L. The Hardy Inequality: About Its History and Some Related Results. Plzen: Vydavatelsky Servis Publishing House, 2007
10	Lane J, Edmunds D. Eigenvalues, Embeddings and Generalised Trigonometric Functions. Lecture Notes in Math, Vol 2016. Berlin: Springer, 2011
11	Opic B, Kufner A. Hardy Type Inequalities. Harlow: Longman Scientific and Technical, 1990
12	Pinasco J. Comparison of eignvalues for the p -Laplacian with integral inequalities. Appl Math Comput, 2006, 182: 1399-1404 https://doi.org/10.1016/j.amc.2006.05.027
13	Wa?ewski T. Sur un principe topologique del’examen de l’allure asymptotique des intégrales deséquations différentielles ordinaires. Ann Soc Polon Math, 1947, 20: 279-313

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