American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2021155

The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian

Fang Liu

Department of Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China

Received  October 2020 Revised  March 2021 Published  June 2021

Fund Project: This work was supported by National Natural Science Foundation of China (No. 11501292)

In this paper, we investigate a weighted Dirichlet eigenvalue problem for a class of degenerate operators related to the
$ h $
 degree homogeneous
$ p $
-Laplacian
$ \begin{equation} \nonumber \left \{ \begin{array}{ll} {|Du|^{h-1}}\Delta_p^N u+ \lambda a(x)|u|^{h-1}u = 0, \quad\quad \rm{in}\quad \Omega, \\ u = 0, \quad\quad \quad \quad \rm{on} \quad\partial\Omega. \end{array}\right. \end{equation} $
Here
$ a(x) $
 is a positive continuous bounded function in the closure of
$ \Omega\subset \mathbb{R}^n(n\geq 2), $
$ h>1, $
$ 2< p<\infty, $
 and
$ \Delta_p^N u = \frac{1}{p}|Du|^{2-p} {\rm div}\left(|Du|^{p-2}Du\right) $
 is the normalized version of the
$ p $
-Laplacian arising from a stochastic game named Tug-of-War with noise. We prove the existence of the principal eigenvalue
$ \lambda_\Omega $
, which is positive and has a corresponding positive eigenfunction for
$ p>n $
. The method is based on the maximum principle and approach analysis to the weighted eigenvalue problem. When a parameter
$ \lambda<\lambda_\Omega $
, we establish some existence and uniqueness results related to this problem. During this procedure, we also prove some regularity estimates including Hölder continuity and Harnack inequality.
Keywords: normalized p-Laplacian, maximum principle, principal eigenvalue, regularity, viscosity solution.
Mathematics Subject Classification: Primary 35J60; 35J70; 35P30.
Citation: Fang Liu. The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021155
References:
[1]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505.  doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[2]

M. Bardi and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math. (Basel), 73 (1999), 276-285.  doi: 10.1007/s000130050399.  Google Scholar

[3]

H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105.  Google Scholar

[4]

H. Berestycki, I.C. Dolcetta, A. Porretta and L. Rossi, Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators, J. Math. Pures Appl., 103 (2015), 1276-1293.  doi: 10.1016/j.matpur.2014.10.012.  Google Scholar

[5]

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

[6]

I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators, Commun. Pure Appl. Anal., 6 (2007), 335-366.  doi: 10.3934/cpaa.2007.6.335.  Google Scholar

[7]

J. Busca, M. J. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 187-206.  doi: 10.1016/j.anihpc.2004.05.004.  Google Scholar

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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.  Google Scholar

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M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations, 13 (2001), 123-139.  doi: 10.1007/s005260000065.  Google Scholar

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K. Does, An evolution equation involving the normalized $p$-Laplacian, Commun. Pure Appl. Anal., 10 (2011), 361-396.  doi: 10.3934/cpaa.2011.10.361.  Google Scholar

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A. Elmoataz, M. Toutain and D. Tenbrinck., On the $p$-Laplacian and $\infty$-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sci., 8 (2015), 2412-2451.  doi: 10.1137/15M1022793.  Google Scholar

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A. Elmoataz, X. Desquesnes and M. Toutain, On the game $p$-Laplacian on weighted graphs with applications in image processing and data clustering, European J. Appl. Math., 28 (2017), 922-948.  doi: 10.1017/S0956792517000122.  Google Scholar

[13]

C. Imbert, T. Jin and L. Silvestre, Hölder gradient estimates for a class of singular or degenerate parabolic equations, Adv. Nonlinear Anal., 8 (2019), 845-867.  doi: 10.1515/anona-2016-0197.  Google Scholar

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H. Ishii, Viscosity solutions of non-linear partial differential equations, Sugaku Expositions, 9 (1996), 135-152.   Google Scholar

[15]

H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78.  doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

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R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal., 123 (1993), 51-74.  doi: 10.1007/BF00386368.  Google Scholar

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P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Differential Equations, 236 (2007), 532-550.  doi: 10.1016/j.jde.2007.01.020.  Google Scholar

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P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal., 33 (2001), 699-717.  doi: 10.1137/S0036141000372179.  Google Scholar

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B. Kawohl, S. Kröemer and J. Kurtz, Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differential Integral Equations, 27 (2014), 659-670.   Google Scholar

[20]

M. Lewicka and J. J. Manfredi, Game theoretical methods in PDEs, Boll. Unione Mat. Ital., 7 (2014), 211-216.  doi: 10.1007/s40574-014-0011-z.  Google Scholar

[21]

F. Liu and F. Jiang, Parabolic biased infinity Laplacian equation related to the biased tug-of-war, Advanced Nonlinear Studies, 19 (2019), 89-112.  doi: 10.1515/ans-2018-2019.  Google Scholar

[22]

Q. Liu and A. Schikorra, General existence of solutions to dynamic programming equations, Commun. Pure Appl. Anal., 14 (2015), 167-184.  doi: 10.3934/cpaa.2015.14.167.  Google Scholar

[23]

F. Liu, L. Tian and P. Zhao, A weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian,, Nonlinear Analysis: TMA, 200 (2020), 112001, 15 pp. doi: 10.1016/j.na.2020.112001.  Google Scholar

[24]

F. Liu and X. Yang, A weighted eigenvalue problem of the biased infinity Laplacian, Nonlinearity, 34 (2021), 1197-1237.  doi: 10.1088/1361-6544/abd85d.  Google Scholar

[25]

G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Part. Diff. Eqns., 33 (2008), 1788-1817.  doi: 10.1080/03605300802289253.  Google Scholar

[26]

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

[27]

J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal., 42 (2010), 2058-2081.  doi: 10.1137/100782073.  Google Scholar

[28]

J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Ann. Scuola Normale Sup. Pisa., 11 (2012), 215-241.   Google Scholar

[29]

J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise, ESAIM Control Optim. Calc. Var., 18 (2012), 81-90.  doi: 10.1051/cocv/2010046.  Google Scholar

[30]

Y. Peres, G. Pete and S. Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones, Calc. Var. Partial Differential Equations, 38 (2010), 541-564.  doi: 10.1007/s00526-009-0298-2.  Google Scholar

[31]

Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1.  Google Scholar

[32]

Y. Peres and S. Sheffield, Tug-of-war with noise: A game theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120.  doi: 10.1215/00127094-2008-048.  Google Scholar

[33]

A. Quaas and B. Sirakov, On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators,, C. R. Math. Acad. Sci. Paris, 342 (2006), 115–118. doi: 10.1016/j.crma.2005.11.003.  Google Scholar

[34]

A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Adv. Math., 218 (2008), 105-135.  doi: 10.1016/j.aim.2007.12.002.  Google Scholar

show all references

References:
[1]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505.  doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[2]

M. Bardi and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math. (Basel), 73 (1999), 276-285.  doi: 10.1007/s000130050399.  Google Scholar

[3]

H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105.  Google Scholar

[4]

H. Berestycki, I.C. Dolcetta, A. Porretta and L. Rossi, Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators, J. Math. Pures Appl., 103 (2015), 1276-1293.  doi: 10.1016/j.matpur.2014.10.012.  Google Scholar

[5]

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

[6]

I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators, Commun. Pure Appl. Anal., 6 (2007), 335-366.  doi: 10.3934/cpaa.2007.6.335.  Google Scholar

[7]

J. Busca, M. J. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 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]

M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations, 13 (2001), 123-139.  doi: 10.1007/s005260000065.  Google Scholar

[10]

K. Does, An evolution equation involving the normalized $p$-Laplacian, Commun. Pure Appl. Anal., 10 (2011), 361-396.  doi: 10.3934/cpaa.2011.10.361.  Google Scholar

[11]

A. Elmoataz, M. Toutain and D. Tenbrinck., On the $p$-Laplacian and $\infty$-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sci., 8 (2015), 2412-2451.  doi: 10.1137/15M1022793.  Google Scholar

[12]

A. Elmoataz, X. Desquesnes and M. Toutain, On the game $p$-Laplacian on weighted graphs with applications in image processing and data clustering, European J. Appl. Math., 28 (2017), 922-948.  doi: 10.1017/S0956792517000122.  Google Scholar

[13]

C. Imbert, T. Jin and L. Silvestre, Hölder gradient estimates for a class of singular or degenerate parabolic equations, Adv. Nonlinear Anal., 8 (2019), 845-867.  doi: 10.1515/anona-2016-0197.  Google Scholar

[14]

H. Ishii, Viscosity solutions of non-linear partial differential equations, Sugaku Expositions, 9 (1996), 135-152.   Google Scholar

[15]

H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78.  doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[16]

R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal., 123 (1993), 51-74.  doi: 10.1007/BF00386368.  Google Scholar

[17]

P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Differential Equations, 236 (2007), 532-550.  doi: 10.1016/j.jde.2007.01.020.  Google Scholar

[18]

P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal., 33 (2001), 699-717.  doi: 10.1137/S0036141000372179.  Google Scholar

[19]

B. Kawohl, S. Kröemer and J. Kurtz, Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differential Integral Equations, 27 (2014), 659-670.   Google Scholar

[20]

M. Lewicka and J. J. Manfredi, Game theoretical methods in PDEs, Boll. Unione Mat. Ital., 7 (2014), 211-216.  doi: 10.1007/s40574-014-0011-z.  Google Scholar

[21]

F. Liu and F. Jiang, Parabolic biased infinity Laplacian equation related to the biased tug-of-war, Advanced Nonlinear Studies, 19 (2019), 89-112.  doi: 10.1515/ans-2018-2019.  Google Scholar

[22]

Q. Liu and A. Schikorra, General existence of solutions to dynamic programming equations, Commun. Pure Appl. Anal., 14 (2015), 167-184.  doi: 10.3934/cpaa.2015.14.167.  Google Scholar

[23]

F. Liu, L. Tian and P. Zhao, A weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian,, Nonlinear Analysis: TMA, 200 (2020), 112001, 15 pp. doi: 10.1016/j.na.2020.112001.  Google Scholar

[24]

F. Liu and X. Yang, A weighted eigenvalue problem of the biased infinity Laplacian, Nonlinearity, 34 (2021), 1197-1237.  doi: 10.1088/1361-6544/abd85d.  Google Scholar

[25]

G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Part. Diff. Eqns., 33 (2008), 1788-1817.  doi: 10.1080/03605300802289253.  Google Scholar

[26]

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

[27]

J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal., 42 (2010), 2058-2081.  doi: 10.1137/100782073.  Google Scholar

[28]

J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Ann. Scuola Normale Sup. Pisa., 11 (2012), 215-241.   Google Scholar

[29]

J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise, ESAIM Control Optim. Calc. Var., 18 (2012), 81-90.  doi: 10.1051/cocv/2010046.  Google Scholar

[30]

Y. Peres, G. Pete and S. Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones, Calc. Var. Partial Differential Equations, 38 (2010), 541-564.  doi: 10.1007/s00526-009-0298-2.  Google Scholar

[31]

Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1.  Google Scholar

[32]

Y. Peres and S. Sheffield, Tug-of-war with noise: A game theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120.  doi: 10.1215/00127094-2008-048.  Google Scholar

[33]

A. Quaas and B. Sirakov, On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators,, C. R. Math. Acad. Sci. Paris, 342 (2006), 115–118. doi: 10.1016/j.crma.2005.11.003.  Google Scholar

[34]

A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear operators, Adv. Math., 218 (2008), 105-135.  doi: 10.1016/j.aim.2007.12.002.  Google Scholar

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