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Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations
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Asymptotic behavior of solutions for nonlinear integral equations with Hénon type on the unit Ball
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 |
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) $.
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. |
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. |
| [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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