-
Previous Article
Dynamical Borel–Cantelli lemmas
- DCDS Home
- This Issue
-
Next Article
Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions
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. |
[1] |
Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021005 |
[2] |
Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254 |
[3] |
Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020445 |
[4] |
Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293 |
[5] |
Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020442 |
[6] |
Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021007 |
[7] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021 doi: 10.3934/nhm.2021003 |
[8] |
Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268 |
[9] |
Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034 |
[10] |
Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020432 |
[11] |
Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control & Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030 |
[12] |
Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020403 |
[13] |
Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040 |
[14] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 |
[15] |
Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020321 |
[16] |
Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088 |
[17] |
Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020469 |
[18] |
Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020379 |
[19] |
Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020466 |
[20] |
Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046 |
2019 Impact Factor: 1.338
Tools
Article outline
[Back to Top]