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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.dam.2007.05.013.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[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), 10 (2004), 28.  doi: 10.1051/cocv:2003035.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1023/A:1024751022715.  Google Scholar

[7]

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

[8]

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

[9]

H. Brezis, Analyse fonctionnelle,, Masson, (1983).   Google Scholar

[10]

J. Bourgain, H. Brezis and P. Mironescu, Another look at sobolev spaces,, in Optimal Control and Partial Differential Equations, (2001), 439.   Google Scholar

[11]

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

[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.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[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.  doi: 10.1002/mana.201200296.  Google Scholar

[14]

F. Demengel and G. Demengel, Functional spaces for the theory of elliptic partial differential equations,, Universitext, (2012).  doi: 10.1007/978-1-4471-2807-6.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2015.04.003.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.  doi: 10.1090/S0002-9947-1991-1083144-2.  Google Scholar

[22]

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

[23]

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

[24]

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

[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.  doi: 10.1007/s13163-014-0147-5.  Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

J. D. Rossi and M. Saez, Optimal regularity for the pseudo infinity Laplacian,, ESAIM. Control, 13 (2007), 294.  doi: 10.1051/cocv:2007018.  Google Scholar

[34]

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

show all references

References:
[1]

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

[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.   Google Scholar

[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.  doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[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), 10 (2004), 28.  doi: 10.1051/cocv:2003035.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1023/A:1024751022715.  Google Scholar

[7]

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

[8]

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

[9]

H. Brezis, Analyse fonctionnelle,, Masson, (1983).   Google Scholar

[10]

J. Bourgain, H. Brezis and P. Mironescu, Another look at sobolev spaces,, in Optimal Control and Partial Differential Equations, (2001), 439.   Google Scholar

[11]

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

[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.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[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.  doi: 10.1002/mana.201200296.  Google Scholar

[14]

F. Demengel and G. Demengel, Functional spaces for the theory of elliptic partial differential equations,, Universitext, (2012).  doi: 10.1007/978-1-4471-2807-6.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2015.04.003.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.  doi: 10.1090/S0002-9947-1991-1083144-2.  Google Scholar

[22]

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

[23]

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

[24]

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

[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.  doi: 10.1007/s13163-014-0147-5.  Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

J. D. Rossi and M. Saez, Optimal regularity for the pseudo infinity Laplacian,, ESAIM. Control, 13 (2007), 294.  doi: 10.1051/cocv:2007018.  Google Scholar

[34]

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

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