• Previous Article
    The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity
  • CPAA Home
  • This Issue
  • Next Article
    Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $
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 $
[1]

Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021006

[2]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018

[3]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[4]

Françoise Demengel. Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021004

[5]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[6]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395

[7]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[8]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[9]

San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038

[10]

Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems & Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067

[11]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180

[12]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012

[13]

Jean-Paul Chehab. Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021002

[14]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[15]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[16]

Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028

[17]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129

[18]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269

[19]

Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031

[20]

Sergey E. Mikhailov, Carlos F. Portillo. Boundary-Domain Integral Equations equivalent to an exterior mixed bvp for the variable-viscosity compressible stokes pdes. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021009

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (75)
  • HTML views (58)
  • Cited by (0)

Other articles
by authors

[Back to Top]