Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions

Mihai Mihăilescu; Julio D. Rossi

doi:10.3934/cpaa.2020198

Article Contents

2020, Volume 19, Issue 9: 4363-4371. Doi: 10.3934/cpaa.2020198

This issue Previous Article Asymptotic behavior of solutions for nonlinear integral equations with Hénon type on the unit Ball Next Article Large time behavior of ODE type solutions to parabolic

$ p $

-Laplacian type equations

Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions

Mihai Mihăilescu^1,2, and
Julio D. Rossi^3, ,

1.
Department of Mathematics, University of Craiova, 200585 Craiova, Romania
2.
Research group of the project PN-III-P1-1.1-TE-2016-2233, University of Bucharest, 010014 Bucharest, Romania
3.
Dep. de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina

^* Corresponding author
^* Corresponding author

Received: August 31, 2019

Revised: February 29, 2020

Published: June 2020

MM was partially supported by CNCS-UEFISCDI Grant No. PN-III-P1-1.1-TE-2016-2233 and JDR by CONICET grant PIP GI No 112201-50100036CO (Argentina) and by UBACyT grant 20020160100155BA (Argentina)

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Abstract

We deal with monotonicity with respect to $ p $ of the first positive eigenvalue of the $ p $-Laplace operator on $ \Omega $ subject to the homogeneous Neumann boundary condition. For any fixed integer $ D>1 $ we show that there exists $ M\in[2 e^{-1}, 2] $ such that for any open, bounded, convex domain $ \Omega\subset{{\mathbb R}}^D $ with smooth boundary for which the diameter of $ \Omega $ is less than or equal to $ M $, the first positive eigenvalue of the $ p $-Laplace operator on $ \Omega $ subject to the homogeneous Neumann boundary condition is an increasing function of $ p $ on $ (1, \infty) $. Moreover, for each real number $ s>M $ there exists a sequence of open, bounded, convex domains $ \{\Omega_n\}_n\subset{{\mathbb R}}^D $ with smooth boundaries for which the sequence of the diameters of $ \Omega_n $ converges to $ s $, as $ n\rightarrow\infty $, and for each $ n $ large enough the first positive eigenvalue of the $ p $-Laplace operator on $ \Omega_n $ subject to the homogeneous Neumann boundary condition is not a monotone function of $ p $ on $ (1, \infty) $.

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Mathematics Subject Classification: Primary: 35P30, 47J10, 49R05; Secondary: 49J40, 58C40.

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References

[1]	S. Azicovici, N. S. Papageorgiu and V. Staicu, The spectrum and index formula for the Neumann $p$-Laplacian and multiple solutions for problems with a crossing nonlinearity, Discrete Contin. Dyn. Syst., 25 (2009), 431-456. doi: 10.3934/dcds.2009.25.431.
[2]	M. Bocea and M. Mihăilescu, On the monotonicity of the principal frequency of the $p$-Laplacian, Adv. Calc. Var., (2019). doi: 10.1515/acv-2018-0022.
[3]	L. Brasco and F. Santambrogio, A note on some Poincaré inequalities on convex sets by optimal transport methods, in Geometric properties for parabolic and elliptic PDE's, Springer Proc. Math. Stat., 176, Springer, [Cham], (2016), 49–63. doi: 10.1007/978-3-319-41538-3_4.
[4]	H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, (2011), xiv+599 pp.
[5]	J. V. Da Silva, J. D. Rossi and A. M. Salort, Uniform stability of the ball with respect to the first Dirichlet and Neumann $\infty$-eiganvalues, Electron. J. Differ. Equ., 7 (2018), 1-9.
[6]	L. Esposito, B. Kawohl, C. Nitsch and C. Trombetti, The Neumann eigenvalue problem for the $\infty$-Laplacian, Rend. Lincei Mat. Appl., 26 (2015), 119-134. doi: 10.4171/RLM/697.
[7]	L. Esposito, C. Nitsch and C. Trombetti, Best constants in Poincaré inequalities for convex domains, J. Conv. Anal., 20 (2013), 253-264.
[8]	N. Fukagai, M. Ito and K. Narukawa, Limit as $p\rightarrow\infty$ of $p$-Laplace eigenvalue problems and $L^\infty$-inequality of Poincaré type, Differ. Integral Equ., 12 (1999), 183-206.
[9]	Y. X. Huang, On the eigenvalues of the $p$-Laplacian with varying $p$, Proc. Amer. Math. Soc., 125 (1997), 3347-3354. doi: 10.1090/S0002-9939-97-03961-0.
[10]	P. Juutinen, P. Lindqvist and J. J. Manfredi, The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105. doi: 10.1007/s002050050157.
[11]	R. Kajikiya, M. Tanaka and S. Tanaka, Bifurcation of positive solutions for the one-dimensional $(p; q)$-Laplace equation, Electron. J. Differ. Equ., 107 (2017), 1-37.
[12]	P. Lindqvist, On non-linear Rayleigh quotients, Potential Anal., 2 (1993), 199-218. doi: 10.1007/BF01048505.
[13]	J. D. Rossi and N. Saintier, On the first nontrivial eigenvalue of the $\infty$-Laplacian with Neumann boundary condition, Huston J. Math., 42 (2016), 613-635.
[14]	D. Valtorta, Sharp estimate on the first eigenvalue of the $p$-Laplacian on compact manifold with nonnegative Ricci curvature, Nonlinear Anal., 75 (2012), 4974-4994. doi: 10.1016/j.na.2012.04.012.