American Institute of Mathematical Sciences

Advanced
  • 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

Pablo Blanc

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.
Keywords: Principal eigenvalue, lower bounds, fully nonlinear elliptic PDEs.
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:
[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.  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. 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.  Google Scholar

[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.  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. Juutinen, P. 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. Kawohl, S. 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-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.  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. 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.  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. 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.  Google Scholar

[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.  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. Juutinen, P. 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. Kawohl, S. 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-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.  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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Functions $ v $ (blue) and $ \phi_{y_0} $ (red) defined in the proof of Theorem 2.2 for a square
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
[2]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795
[3]

Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315
[4]

Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845
[5]

Rainer Buckdahn, Christian Keller, Jin Ma, Jianfeng Zhang. Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 7-. doi: 10.1186/s41546-020-00049-8
[6]

Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081
[7]

Qilong Zhai, Ran Zhang. Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 403-413. doi: 10.3934/dcdsb.2018091
[8]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[9]

Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048
[10]

Olivier Guibé, Anna Mercaldo. Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Communications on Pure & Applied Analysis, 2008, 7 (1) : 163-192. doi: 10.3934/cpaa.2008.7.163
[11]

Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2089-2104. doi: 10.3934/cpaa.2017103
[12]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[13]

Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213
[14]

Luisa Fattorusso, Antonio Tarsia. Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1307-1323. doi: 10.3934/dcds.2011.31.1307
[15]

Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006
[16]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61
[17]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247
[18]

Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019
[19]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187
[20]

Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (115)
  • HTML views (60)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences