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

Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems

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  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.

Citation: Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete & 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.  Google Scholar

[2]

L. BrascoE. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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L. M. Del PezzoJ. 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.  Google Scholar

[6]

E. Di NezzaG. 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.  Google Scholar

[7]

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

[8]

J. Fernández BonderA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

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.  Google Scholar

[2]

L. BrascoE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. M. Del PezzoJ. 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.  Google Scholar

[6]

E. Di NezzaG. 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.  Google Scholar

[7]

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

[8]

J. Fernández BonderA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

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