December  2016, 36(12): 6737-6765. doi: 10.3934/dcds.2016093

Eigenvalues for a nonlocal pseudo $p-$Laplacian

1. 

CONICET and Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria, Buenos Aires, 1428, Argentina, Argentina

Received  November 2015 Revised  August 2016 Published  October 2016

In this paper we study the eigenvalue problems for a nonlocal operator of order $s$ that is analogous to the local pseudo $p-$Laplacian. We show that there is a sequence of eigenvalues $\lambda_n \to \infty$ and that the first one is positive, simple, isolated and has a positive and bounded associated eigenfunction. For the first eigenvalue we also analyze the limits as $p\to \infty$ (obtaining a limit nonlocal eigenvalue problem analogous to the pseudo infinity Laplacian) and as $s\to 1^-$ (obtaining the first eigenvalue for a local operator of $p-$Laplacian type). To perform this study we have to introduce anisotropic fractional Sobolev spaces and prove some of their properties.
Citation: Leandro M. Del Pezzo, Julio D. Rossi. Eigenvalues for a nonlocal pseudo $p-$Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6737-6765. doi: 10.3934/dcds.2016093
References:
[1]

S. Amghibech, On the discrete version of Picone's identity, Discrete Appl. Math., 156 (2008), 1-10. doi: 10.1016/j.dam.2007.05.013.

[2]

A. Anane, Simplicité et isolation de la première valeur propre du $p-$laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725-728.

[3]

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.

[4]

M. Belloni and B. Kawohl, The pseudo-p-Laplace eigenvalue problem and viscosity solutions as $p\to \infty$, ESAIM Control Optim. Calc. Var. 10 (2004), 28-52. doi: 10.1051/cocv:2003035.

[5]

T. Bhattacharya, E. Di Benedetto and J. Manfredi, Limits as $p \to \infty$ of $\Delta_p u_p = f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1991), 15-68.

[6]

G. Bouchitte, G. Buttazzo and L. De Pasquale., A $p-$laplacian approximation for some mass optimization problems, J. Optim. Theory Appl., 118 (2003), 1-25. doi: 10.1023/A:1024751022715.

[7]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799. doi: 10.2996/kmj/1414674621.

[8]

L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p-$Laplacian, Discr. Cont. Dyn. Sys., 36 (2016), 1813-1845. doi: 10.3934/dcds.2016.36.1813.

[9]

H. Brezis, Analyse fonctionnelle, Masson, Paris, 1983.

[10]

J. Bourgain, H. Brezis and P. Mironescu, Another look at sobolev spaces, in Optimal Control and Partial Differential Equations, 2001, 439-455.

[11]

A. Chambolle, E. Lindgren and R. Monneau, The Holder infinite Laplacian and Holder extensions, ESAIM-COCV, 18 (2012), 799-835. doi: 10.1051/cocv/2011182.

[12]

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., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.

[13]

F. Della Pietra and N. Gavitone, Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators, Math. Nachr., 287 (2014), 194-209. doi: 10.1002/mana.201200296.

[14]

F. Demengel and G. Demengel, Functional spaces for the theory of elliptic partial differential equations, Universitext, Springer, London, 2012, Translated from the 2007 French original by Reinie Erné. doi: 10.1007/978-1-4471-2807-6.

[15]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299. doi: 10.1016/j.anihpc.2015.04.003.

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[17]

S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, to appear in Rev. Mat. Iberoam..

[18]

L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc., 137 (1999), viii+66 pp. doi: 10.1090/memo/0653.

[19]

L. C. Evans and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations, 42 (2011), 289-299. doi: 10.1007/s00526-010-0388-1.

[20]

L. C. Evans and C. K. Smart, Adjoint methods for the infinity Laplacian partial differential equation, Arch. Ration. Mech. Anal., 201 (2011), 87-113. doi: 10.1007/s00205-011-0399-x.

[21]

J. Garcia-Azorero and I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895. doi: 10.1090/S0002-9947-1991-1083144-2.

[22]

G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.

[23]

A. Iannizzotto, S. Mosconi, and M. Squassina, Global Hölder regularity for the fractional $p-$Laplacian, to appear in Rev. Mat. Iberoam.

[24]

J. Jaros, Picone's identity for a Finsler p-Laplacian and comparison of nonlinear elliptic equations, Math. Bohem.,139 (2014), 535-552.

[25]

H. Jylha, An optimal transportation problem related to the limits of solutions of local and nonlocal $p-$Laplace- type problems, Rev. Mat. Complutense, 28 (2015), 85-121. doi: 10.1007/s13163-014-0147-5.

[26]

P. Juutinen, P. Lindqvist and J. J. Manfredi, The $\infty-$eigenvalue problem, Arch. Rational Mech. Anal., 148 (1999), 89-105. doi: 10.1007/s002050050157.

[27]

P. Juutinen and P. Lindqvist, On the higher eigenvalues for the $\infty-$eigenvalue problem, Calc. Var. Partial Differential Equations, 23 (2005), 169-192. doi: 10.1007/s00526-004-0295-4.

[28]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826. doi: 10.1007/s00526-013-0600-1.

[29]

G. Molica Bisci, Sequence of weak solutions for fractional equations, Math. Research Lett., 21 (2014), 241-253. doi: 10.4310/MRL.2014.v21.n2.a3.

[30]

G. Molica Bisci, Fractional equations with bounded primitive, Appl. Math. Lett., 27 (2014), 53-58. doi: 10.1016/j.aml.2013.07.011.

[31]

G. Molica Bisci and B. A. Pansera, Three weak solutions for nonlocal fractional equations, Adv. Nonlinear Stud., 14 (2014), 619-629.

[32]

M. Moussa, Schwarz rearrangement does not decrease the energy for the pseudo $p-$Laplacian operator, Bol. Soc. Parana. Mat. (3), 29 (2011), 49-53. doi: 10.5269/bspm.v29i1.10428.

[33]

J. D. Rossi and M. Saez, Optimal regularity for the pseudo infinity Laplacian, ESAIM. Control, Opt. Calc. Var., COCV., 13 (2007), 294-304. doi: 10.1051/cocv:2007018.

[34]

O. Savin, $C^1$ regularity for infinity harmonic functions in two dimensions, Arch. Rational Mech. Anal., 176 (2005), 351-361. doi: 10.1007/s00205-005-0355-8.

show all references

References:
[1]

S. Amghibech, On the discrete version of Picone's identity, Discrete Appl. Math., 156 (2008), 1-10. doi: 10.1016/j.dam.2007.05.013.

[2]

A. Anane, Simplicité et isolation de la première valeur propre du $p-$laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725-728.

[3]

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.

[4]

M. Belloni and B. Kawohl, The pseudo-p-Laplace eigenvalue problem and viscosity solutions as $p\to \infty$, ESAIM Control Optim. Calc. Var. 10 (2004), 28-52. doi: 10.1051/cocv:2003035.

[5]

T. Bhattacharya, E. Di Benedetto and J. Manfredi, Limits as $p \to \infty$ of $\Delta_p u_p = f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1991), 15-68.

[6]

G. Bouchitte, G. Buttazzo and L. De Pasquale., A $p-$laplacian approximation for some mass optimization problems, J. Optim. Theory Appl., 118 (2003), 1-25. doi: 10.1023/A:1024751022715.

[7]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799. doi: 10.2996/kmj/1414674621.

[8]

L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p-$Laplacian, Discr. Cont. Dyn. Sys., 36 (2016), 1813-1845. doi: 10.3934/dcds.2016.36.1813.

[9]

H. Brezis, Analyse fonctionnelle, Masson, Paris, 1983.

[10]

J. Bourgain, H. Brezis and P. Mironescu, Another look at sobolev spaces, in Optimal Control and Partial Differential Equations, 2001, 439-455.

[11]

A. Chambolle, E. Lindgren and R. Monneau, The Holder infinite Laplacian and Holder extensions, ESAIM-COCV, 18 (2012), 799-835. doi: 10.1051/cocv/2011182.

[12]

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., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.

[13]

F. Della Pietra and N. Gavitone, Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators, Math. Nachr., 287 (2014), 194-209. doi: 10.1002/mana.201200296.

[14]

F. Demengel and G. Demengel, Functional spaces for the theory of elliptic partial differential equations, Universitext, Springer, London, 2012, Translated from the 2007 French original by Reinie Erné. doi: 10.1007/978-1-4471-2807-6.

[15]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299. doi: 10.1016/j.anihpc.2015.04.003.

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[17]

S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, to appear in Rev. Mat. Iberoam..

[18]

L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc., 137 (1999), viii+66 pp. doi: 10.1090/memo/0653.

[19]

L. C. Evans and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations, 42 (2011), 289-299. doi: 10.1007/s00526-010-0388-1.

[20]

L. C. Evans and C. K. Smart, Adjoint methods for the infinity Laplacian partial differential equation, Arch. Ration. Mech. Anal., 201 (2011), 87-113. doi: 10.1007/s00205-011-0399-x.

[21]

J. Garcia-Azorero and I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895. doi: 10.1090/S0002-9947-1991-1083144-2.

[22]

G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.

[23]

A. Iannizzotto, S. Mosconi, and M. Squassina, Global Hölder regularity for the fractional $p-$Laplacian, to appear in Rev. Mat. Iberoam.

[24]

J. Jaros, Picone's identity for a Finsler p-Laplacian and comparison of nonlinear elliptic equations, Math. Bohem.,139 (2014), 535-552.

[25]

H. Jylha, An optimal transportation problem related to the limits of solutions of local and nonlocal $p-$Laplace- type problems, Rev. Mat. Complutense, 28 (2015), 85-121. doi: 10.1007/s13163-014-0147-5.

[26]

P. Juutinen, P. Lindqvist and J. J. Manfredi, The $\infty-$eigenvalue problem, Arch. Rational Mech. Anal., 148 (1999), 89-105. doi: 10.1007/s002050050157.

[27]

P. Juutinen and P. Lindqvist, On the higher eigenvalues for the $\infty-$eigenvalue problem, Calc. Var. Partial Differential Equations, 23 (2005), 169-192. doi: 10.1007/s00526-004-0295-4.

[28]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826. doi: 10.1007/s00526-013-0600-1.

[29]

G. Molica Bisci, Sequence of weak solutions for fractional equations, Math. Research Lett., 21 (2014), 241-253. doi: 10.4310/MRL.2014.v21.n2.a3.

[30]

G. Molica Bisci, Fractional equations with bounded primitive, Appl. Math. Lett., 27 (2014), 53-58. doi: 10.1016/j.aml.2013.07.011.

[31]

G. Molica Bisci and B. A. Pansera, Three weak solutions for nonlocal fractional equations, Adv. Nonlinear Stud., 14 (2014), 619-629.

[32]

M. Moussa, Schwarz rearrangement does not decrease the energy for the pseudo $p-$Laplacian operator, Bol. Soc. Parana. Mat. (3), 29 (2011), 49-53. doi: 10.5269/bspm.v29i1.10428.

[33]

J. D. Rossi and M. Saez, Optimal regularity for the pseudo infinity Laplacian, ESAIM. Control, Opt. Calc. Var., COCV., 13 (2007), 294-304. doi: 10.1051/cocv:2007018.

[34]

O. Savin, $C^1$ regularity for infinity harmonic functions in two dimensions, Arch. Rational Mech. Anal., 176 (2005), 351-361. doi: 10.1007/s00205-005-0355-8.

[1]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019

[2]

Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355

[3]

Huicong Li, Jingyu Li. Asymptotic behavior of Dirichlet eigenvalues on a body coated by functionally graded material. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1493-1516. doi: 10.3934/cpaa.2017071

[4]

Kehan Shi, Ying Wen. Nonlocal biharmonic evolution equations with Dirichlet and Navier boundary conditions. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022089

[5]

Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure and Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689

[6]

Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure and Applied Analysis, 2005, 4 (3) : 683-693. doi: 10.3934/cpaa.2005.4.683

[7]

Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441

[8]

E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39

[9]

Vesselin Petkov. Location of eigenvalues for the wave equation with dissipative boundary conditions. Inverse Problems and Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034

[10]

Monica Conti, Stefania Gatti, Alain Miranville. Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 485-505. doi: 10.3934/dcdss.2012.5.485

[11]

Ciprian G. Gal, Hao Wu. Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 1041-1063. doi: 10.3934/dcds.2008.22.1041

[12]

Abdessatar Khelifi, Siwar Saidani. Asymptotic behavior of eigenvalues of the Maxwell system in the presence of small changes in the interface of an inclusion. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022080

[13]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295

[14]

Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391

[15]

Micol Amar. A note on boundary layer effects in periodic homogenization with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 537-556. doi: 10.3934/dcds.2000.6.537

[16]

Gennaro Infante. Eigenvalues and positive solutions of odes involving integral boundary conditions. Conference Publications, 2005, 2005 (Special) : 436-442. doi: 10.3934/proc.2005.2005.436

[17]

Guangdong Jing, Penghui Wang. Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021055

[18]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095

[19]

Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041

[20]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]