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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

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

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) $.

Keywords: Diameter, open, bounded, convex domain, first positive eigenvalue, p-Laplacian, Neumann boundary condition.
Mathematics Subject Classification: Primary: 35P30, 47J10, 49R05; Secondary: 49J40, 58C40.
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. 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.  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

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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 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.   Google Scholar

[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.  Google Scholar

[7]

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

[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.   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. Juutinen, P. 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. 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.   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. 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.  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 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.   Google Scholar

[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.  Google Scholar

[7]

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

[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.   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. Juutinen, P. 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. 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.   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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