September  2020, 19(9): 4363-4371. doi: 10.3934/cpaa.2020198

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

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

Received  September 2019 Revised  March 2020 Published  June 2020

Fund Project: 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)

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) $.

Citation: 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
References:
[1]

S. AzicoviciN. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, (2011), xiv+599 pp.  Google Scholar

[5]

J. V. Da SilvaJ. 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.   Google Scholar

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L. EspositoB. KawohlC. 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.  Google Scholar

[7]

L. EspositoC. Nitsch and C. Trombetti, Best constants in Poincaré inequalities for convex domains, J. Conv. Anal., 20 (2013), 253-264.   Google Scholar

[8]

N. FukagaiM. 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.   Google Scholar

[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.  Google Scholar

[10]

P. JuutinenP. Lindqvist and J. J. Manfredi, The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105.  doi: 10.1007/s002050050157.  Google Scholar

[11]

R. KajikiyaM. Tanaka and S. Tanaka, Bifurcation of positive solutions for the one-dimensional $(p; q)$-Laplace equation, Electron. J. Differ. Equ., 107 (2017), 1-37.   Google Scholar

[12]

P. Lindqvist, On non-linear Rayleigh quotients, Potential Anal., 2 (1993), 199-218.  doi: 10.1007/BF01048505.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. AzicoviciN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, (2011), xiv+599 pp.  Google Scholar

[5]

J. V. Da SilvaJ. 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.   Google Scholar

[6]

L. EspositoB. KawohlC. 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.  Google Scholar

[7]

L. EspositoC. Nitsch and C. Trombetti, Best constants in Poincaré inequalities for convex domains, J. Conv. Anal., 20 (2013), 253-264.   Google Scholar

[8]

N. FukagaiM. 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.   Google Scholar

[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.  Google Scholar

[10]

P. JuutinenP. Lindqvist and J. J. Manfredi, The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105.  doi: 10.1007/s002050050157.  Google Scholar

[11]

R. KajikiyaM. Tanaka and S. Tanaka, Bifurcation of positive solutions for the one-dimensional $(p; q)$-Laplace equation, Electron. J. Differ. Equ., 107 (2017), 1-37.   Google Scholar

[12]

P. Lindqvist, On non-linear Rayleigh quotients, Potential Anal., 2 (1993), 199-218.  doi: 10.1007/BF01048505.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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