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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.

Received  June 2019 Revised  January 2020 Published  April 2020

Fund Project: This work has been partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-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 $
is the largest radius of a ball included in the domain
$ \Omega\subset {\mathbb R}^n $
, and
$ \lambda_{1,p}(\Omega) $
and
$ \lambda_{1,\infty}(\Omega) $
are the principal eigenvalue for the homogeneous
$ p $
-laplacian and the homogeneous infinity laplacian respectively.
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:
[1]

H. BerestyckiL. 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.  Google Scholar

[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.  Google Scholar

[3]

I. Birindelli and F. Demengel, First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differ. Equ., 11 (2006), 91-119.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. BuscaM. 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.  Google Scholar

[8]

M. G. CrandallH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

B. KawohlS. Krömer and J. Kurtz, Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differ. Integral Equ., 27 (2014), 659-670.   Google Scholar

[14]

P. J. Martínez-AparicioM. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. BerestyckiL. 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.  Google Scholar

[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.  Google Scholar

[3]

I. Birindelli and F. Demengel, First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differ. Equ., 11 (2006), 91-119.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. BuscaM. 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.  Google Scholar

[8]

M. G. CrandallH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

B. KawohlS. Krömer and J. Kurtz, Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differ. Integral Equ., 27 (2014), 659-670.   Google Scholar

[14]

P. J. Martínez-AparicioM. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  Radial functions that allow us to obtain bounds for the principal eigenvalue
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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