American Institute of Mathematical Sciences

Advanced
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  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 & Continuous Dynamical Systems - A, 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. 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.  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 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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

[1]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023
[2]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133
[3]

Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068
[4]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021
[5]

Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021004
[6]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[7]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035
[8]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[9]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[10]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022
[11]

Zhimin Chen, Kaihui Liu, Xiuxiang Liu. Evaluating vaccination effectiveness of group-specific fractional-dose strategies. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021062
[12]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450
[13]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066
[14]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[15]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[16]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019
[17]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017
[18]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020
[19]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[20]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

2019 Impact Factor: 1.338

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences