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Minimal period solutions in asymptotically linear Hamiltonian system with symmetries
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 |
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.
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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