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doi: 10.3934/dcdsb.2021155
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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 Early access 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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