# American Institute of Mathematical Sciences

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 and 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. [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.

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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.
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