Figure 1.
Radial functions that allow us to obtain bounds for the principal eigenvalue
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July
2020, 19(7): 3613-3623.
doi: 10.3934/cpaa.2020158
A lower bound for the principal eigenvalue of fully nonlinear elliptic operators
Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina. |
In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We illustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that
| $ \lim_{p\to \infty}\lambda_{1,p}(\Omega) = \lambda_{1,\infty}(\Omega) = \left(\frac{\pi}{2R}\right)^2 $ |
where
| $ R $ |
| $ \Omega\subset {\mathbb R}^n $ |
| $ \lambda_{1,p}(\Omega) $ |
| $ \lambda_{1,\infty}(\Omega) $ |
| $ p $ |
Mathematics Subject Classification:
Primary: 35P15; Secondary: 35P30, 35J60, 35J70.
Citation:
Pablo Blanc. A lower bound for the principal eigenvalue of fully nonlinear elliptic operators. Communications on Pure & Applied Analysis,
2020, 19
(7)
: 3613-3623.
doi: 10.3934/cpaa.2020158
![]()
References:
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H. Berestycki, L. Nirenberg and S. R. S. Varadhan,
The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Commun. Pure Appl. Math., 47 (1994), 47-92.
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H. Berestycki and L. Rossi,
Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Commun. Pure Appl. Math., 68 (2015), 1014-1065.
doi: 10.1002/cpa.21536. |
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I. Birindelli and F. Demengel,
First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differ. Equ., 11 (2006), 91-119.
|
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I. Birindelli and F. Demengel,
Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math. (6), 13 (2004), 261-287.
|
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I. Birindelli and F. Demengel,
Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains, J. Math. Anal. Appl., 352 (2009), 822-835.
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I. Birindelli and F. Demengel,
Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differ. Equ., 249 (2010), 1089-1110.
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J. Busca, M. J. Esteban and A. Quaas,
Nonlinear eigenvalues and bifurcation problems for {P}ucci's operators, Ann. Inst. Henri Poincare Anal. Non Lineaire, 22 (2005), 187-206.
doi: 10.1016/j.anihpc.2004.05.004. |
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M. G. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. N.S., 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
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G. Crasta and I. Fragalà,
Characterization of stadium-like domains via boundary value problems for the infinity Laplacian, Nonlinear Anal., 133 (2016), 228-249.
doi: 10.1016/j.na.2015.12.007. |
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P. Juutinen,
Principal eigenvalue of a very badly degenerate operator and applications, J. Differ. Equ., 236 (2007), 532-550.
doi: 10.1016/j.jde.2007.01.020. |
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P. Juutinen, P. Lindqvist and J. J. Manfredi,
The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105.
doi: 10.1007/s002050050157. |
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B. Kawohl and J. Horák,
On the geometry of the $p$-Laplacian operator, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 799-813.
doi: 10.3934/dcdss.2017040. |
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B. Kawohl, S. Krömer and J. Kurtz,
Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differ. Integral Equ., 27 (2014), 659-670.
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P. J. Martínez-Aparicio, M. Pérez-Llanos and J. D. Rossi,
The limit as $p\rightarrow\infty$ for the eigenvalue problem of the 1-homogeneous $p$-Laplacian, Rev. Mat. Complut., 27 (2014), 241-258.
doi: 10.1007/s13163-013-0124-4. |
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A. Quaas and B. Sirakov,
Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105-135.
doi: 10.1016/j.aim.2007.12.002. |
show all references
References:
| [1] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan,
The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Commun. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
| [2] |
H. Berestycki and L. Rossi,
Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Commun. Pure Appl. Math., 68 (2015), 1014-1065.
doi: 10.1002/cpa.21536. |
| [3] |
I. Birindelli and F. Demengel,
First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differ. Equ., 11 (2006), 91-119.
|
| [4] |
I. Birindelli and F. Demengel,
Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math. (6), 13 (2004), 261-287.
|
| [5] |
I. Birindelli and F. Demengel,
Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains, J. Math. Anal. Appl., 352 (2009), 822-835.
doi: 10.1016/j.jmaa.2008.11.012. |
| [6] |
I. Birindelli and F. Demengel,
Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differ. Equ., 249 (2010), 1089-1110.
doi: 10.1016/j.jde.2010.03.015. |
| [7] |
J. Busca, M. J. Esteban and A. Quaas,
Nonlinear eigenvalues and bifurcation problems for {P}ucci's operators, Ann. Inst. Henri Poincare Anal. Non Lineaire, 22 (2005), 187-206.
doi: 10.1016/j.anihpc.2004.05.004. |
| [8] |
M. G. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. N.S., 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
| [9] |
G. Crasta and I. Fragalà,
Characterization of stadium-like domains via boundary value problems for the infinity Laplacian, Nonlinear Anal., 133 (2016), 228-249.
doi: 10.1016/j.na.2015.12.007. |
| [10] |
P. Juutinen,
Principal eigenvalue of a very badly degenerate operator and applications, J. Differ. Equ., 236 (2007), 532-550.
doi: 10.1016/j.jde.2007.01.020. |
| [11] |
P. Juutinen, P. Lindqvist and J. J. Manfredi,
The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105.
doi: 10.1007/s002050050157. |
| [12] |
B. Kawohl and J. Horák,
On the geometry of the $p$-Laplacian operator, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 799-813.
doi: 10.3934/dcdss.2017040. |
| [13] |
B. Kawohl, S. Krömer and J. Kurtz,
Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differ. Integral Equ., 27 (2014), 659-670.
|
| [14] |
P. J. Martínez-Aparicio, M. Pérez-Llanos and J. D. Rossi,
The limit as $p\rightarrow\infty$ for the eigenvalue problem of the 1-homogeneous $p$-Laplacian, Rev. Mat. Complut., 27 (2014), 241-258.
doi: 10.1007/s13163-013-0124-4. |
| [15] |
A. Quaas and B. Sirakov,
Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105-135.
doi: 10.1016/j.aim.2007.12.002. |
Figure 2.
In black an L shaped domain, in red the ball of maximum radius contained in the domain, in green the ball of minimum radius that contains the domain and in blue the boundary of the narrowest strip that contains the domain
Figure 3.
Functions $ v $ (blue) and $ \phi_{y_0} $ (red) defined in the proof of Theorem 2.2 for a square
Figure 4.
In black a U shaped domain ($ \Omega $ ), in red the ball of maximum radius ($ R $ ) included in $ \Omega $ , in blue $ \Omega_\delta $ and in green the ball of maximum radius ($ R_\delta $ ) included in $ \Omega_\delta $ , we have $ R_\delta>R+\delta $
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