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May  2021, 41(5): 2125-2140. doi: 10.3934/dcds.2020355

Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems

Julián Fernández Bonder 1,, , Analía Silva 2, and  Juan F. Spedaletti 2,

1. 

Instituto de Matemática Luis A. Santaló (IMAS), CONICET, Departamento de Matemática, FCEN - Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, C1428EGA, Av. Cantilo s/n, Buenos Aires, Argentina
2. 

Instituto de Matemática Aplicada San luis (IMASL), Ejército de los Andes 950, D5700HHW, San Luis, Argentina

* Corresponding author: J. Fernández Bonder

Received  January 2020 Revised  September 2020 Published  May 2021 Early access  October 2020

In this paper we analyze the asymptotic behavior of several fractional eigenvalue problems by means of Gamma-convergence methods. This method allows us to treat different eigenvalue problems under a unified framework. We are able to recover some known results for the behavior of the eigenvalues of the $ p- $fractional laplacian when the fractional parameter $ s $ goes to 1, and to extend some known results for the behavior of the same eigenvalue problem when $ p $ goes to $ \infty $. Finally we analyze other eigenvalue problems not previously covered in the literature.

Keywords: Fractional eigenvalues, stability of nonlinear eigenvalues, fractional p-laplacian problems.
Mathematics Subject Classification: Primary: 35P30, 35J92; Secondary: 49R05.
Citation: 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
References:
[1]

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

[2]

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

[3]

T. Champion and L. De Pascale, Asymptotic behaviour of nonlinear eigenvalue problems involving $p-$laplacian-type operators, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1179-1195.  doi: 10.1017/S0308210506000667.

[4]

G. Dal Maso, An Introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[5]

L. M. Del Pezzo, J. D. Rossi and A. M. Salort, Fractional eigenvalue problems that approximate steklov eigenvalue problems, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 148 (2018), 499-516.  doi: 10.1017/S0308210517000361.

[6]

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.

[7]

J. Fernández Bonder, J. P. Pinasco and A. M. Salort, Eigenvalue homogenisation problem with indefinite weights, Bull. Aust. Math. Soc., 93 (2016), 113-127.  doi: 10.1017/S0004972715001094.

[8]

J. Fernández Bonder, A. Ritorto and A. M. Salort, $H$-convergence result for nonlocal elliptic-type problems via Tartar's method, SIAM J. Math. Anal., 49 (2017), 2387-2408.  doi: 10.1137/16M1080215.

[9]

J. Fernández Bonder and A. M. Salort, Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., 277 (2019), 333-367.  doi: 10.1016/j.jfa.2019.04.003.

[10]

J. Fernández Bonder and A. M. Salort, Stability of solutions for nonlocal problems, Nonlinear Analysis, 200 (2020), 112080, 13 pp. doi: 10.1016/j.na.2020.112080.

[11]

M. Focardi, Aperiodic fractional obstacle problems, Adv. Math., 225 (2010), 3502-3544.  doi: 10.1016/j.aim.2010.06.014.

[12]

_____, Γ-convergence: A tool to investigate physical phenomena across scales, Math. Methods Appl. Sci., 35 (2012), 1613-1658. doi: 10.1002/mma.2551.

[13]

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

[14]

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

[15]

A. Piatnitski and E. Zhizhina, Periodic homogenization of nonlocal operators with a convolution-type kernel, SIAM J. Math. Anal., 49 (2017), 64-81.  doi: 10.1137/16M1072292.

[16]

A. C. Ponce, A new approach to Sobolev spaces and connections to $\Gamma$-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z.

[17]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.

[18]

R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680.  doi: 10.1137/080737897.

[19]

_____, Stochastic homogenization for some nonlinear integro-differential equations, Comm. Partial Differential Equations, 38 (2013), 171-198. doi: 10.1080/03605302.2012.741176.

show all references

References:
[1]

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

[2]

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

[3]

T. Champion and L. De Pascale, Asymptotic behaviour of nonlinear eigenvalue problems involving $p-$laplacian-type operators, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1179-1195.  doi: 10.1017/S0308210506000667.

[4]

G. Dal Maso, An Introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[5]

L. M. Del Pezzo, J. D. Rossi and A. M. Salort, Fractional eigenvalue problems that approximate steklov eigenvalue problems, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 148 (2018), 499-516.  doi: 10.1017/S0308210517000361.

[6]

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.

[7]

J. Fernández Bonder, J. P. Pinasco and A. M. Salort, Eigenvalue homogenisation problem with indefinite weights, Bull. Aust. Math. Soc., 93 (2016), 113-127.  doi: 10.1017/S0004972715001094.

[8]

J. Fernández Bonder, A. Ritorto and A. M. Salort, $H$-convergence result for nonlocal elliptic-type problems via Tartar's method, SIAM J. Math. Anal., 49 (2017), 2387-2408.  doi: 10.1137/16M1080215.

[9]

J. Fernández Bonder and A. M. Salort, Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., 277 (2019), 333-367.  doi: 10.1016/j.jfa.2019.04.003.

[10]

J. Fernández Bonder and A. M. Salort, Stability of solutions for nonlocal problems, Nonlinear Analysis, 200 (2020), 112080, 13 pp. doi: 10.1016/j.na.2020.112080.

[11]

M. Focardi, Aperiodic fractional obstacle problems, Adv. Math., 225 (2010), 3502-3544.  doi: 10.1016/j.aim.2010.06.014.

[12]

_____, Γ-convergence: A tool to investigate physical phenomena across scales, Math. Methods Appl. Sci., 35 (2012), 1613-1658. doi: 10.1002/mma.2551.

[13]

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

[14]

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

[15]

A. Piatnitski and E. Zhizhina, Periodic homogenization of nonlocal operators with a convolution-type kernel, SIAM J. Math. Anal., 49 (2017), 64-81.  doi: 10.1137/16M1072292.

[16]

A. C. Ponce, A new approach to Sobolev spaces and connections to $\Gamma$-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.  doi: 10.1007/s00526-003-0195-z.

[17]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.

[18]

R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680.  doi: 10.1137/080737897.

[19]

_____, Stochastic homogenization for some nonlinear integro-differential equations, Comm. Partial Differential Equations, 38 (2013), 171-198. doi: 10.1080/03605302.2012.741176.

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