Stability of variational eigenvalues
for the fractional $p-$Laplacian

Lorenzo  Brasco; Enea  Parini; Marco Squassina

doi:10.3934/dcds.2016.36.1813

Previous Article
Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows
DCDS Home
This Issue
Next Article
Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations

April 2016, 36(4): 1813-1845. doi: 10.3934/dcds.2016.36.1813

Stability of variational eigenvalues for the fractional $p-$Laplacian

Lorenzo Brasco ^1, , Enea Parini ^1, and Marco Squassina ^2,

1.	Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 Rue Frédéric Joliot Curie, 13453 Marseille, France, France
2.	Dipartimento di Informatica, Università di Verona, Strada Le Grazie 15, 37134 Verona

Received March 2015 Revised May 2015 Published September 2015

Abstract
Related Papers

By virtue of $\Gamma-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p-$Laplacian operator, in the singular limit as the nonlocal operator converges to the $p-$Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.

Keywords: $\Gamma-$convergence., nonlocal eigenvalue problems, Fractional $p-$Laplacian, critical points.

Mathematics Subject Classification: 35P30, 49J35, 49J4.

Citation: Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813

References:

[1]	L. Ambrosio, G. De Philippis and L. Martinazzi, $\Gamma$-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403. doi: 10.1007/s00229-010-0399-4. Google Scholar
[2]	D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. Google Scholar
[3]	J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan's 60th Birthday (eds. J. L. Menaldi, E. Rofman and A. Sulem), IOS Press, Amsterdam, 2001, 439-455. Google Scholar
[4]	J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s \to 1$ and applications, J. Anal. Math., 87 (2002), 77-101. doi: 10.1007/BF02868470. Google Scholar
[5]	H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983. Google Scholar
[6]	L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458. doi: 10.4171/IFB/325. Google Scholar
[7]	L. Brasco and E. Parini, The second eigenvalue of the fractional $p$-Laplacian, to appear on Adv. Calc. Var. (2015) doi: 10.1515/acv-2015-0007. Google Scholar
[8]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar
[9]	L. A. Caffarelli, Nonlocal equations, drifts and games, Nonlinear partial differential equations, Abel Symposia, 7 (2012), 37-52. doi: 10.1007/978-3-642-25361-4_3. Google Scholar
[10]	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
[11]	M. Cuesta, D. G. De Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian, J. Differential Equations, 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645. Google Scholar
[12]	G. Dal Maso, An Introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8. Google Scholar
[13]	M. Degiovanni and M. Marzocchi, Limit of minimax values under $\Gamma$-convergence, Electron. J. Differential Equations, (2014), 19pp. Google Scholar
[14]	A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, to appear on Ann. Inst. H. Poincaré Anal. Non Linéaire (2015) doi: 10.1016/j.anihpc.2015.04.003. Google Scholar
[15]	B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588. Google Scholar
[16]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. Google Scholar
[17]	E. R. Fadell and P. H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal., 26 (1977), 48-67. doi: 10.1016/0022-1236(77)90015-5. Google Scholar
[18]	A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253. doi: 10.5186/aasfm.2015.4009. Google Scholar
[19]	I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics. Springer, New York, 2007. Google Scholar
[20]	I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in $L^1$, SIAM J. Math. Anal., 23 (1992), 1081-1098. doi: 10.1137/0523060. Google Scholar
[21]	G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. Google Scholar
[22]	E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557. Google Scholar
[23]	A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006. Google Scholar
[24]	Y. X. Huang, On the eigenvalue of the $p$-Laplacian with varying $p$, Proc. Amer. Math. Soc., 125 (1997), 3347-3354. doi: 10.1090/S0002-9939-97-03961-0. Google Scholar
[25]	A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, to appear on Adv. Calc. Var., (2015) doi: 10.1515/acv-2014-0024. Google Scholar
[26]	A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, to appear on Rev. Mat. Iberoam.,, available at , (). Google Scholar
[27]	A. Iannizzotto and M. Squassina, Weyl-type laws for fractional $p$-eigenvalue problems, Asymptot. Anal., 88 (2014), 233-245. Google Scholar
[28]	T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368. doi: 10.1007/s00220-015-2356-2. Google Scholar
[29]	T. Kuusi, G. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57-114. doi: 10.2140/apde.2015.8.57. Google Scholar
[30]	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
[31]	P. Lindqvist, On non-linear Rayleigh quotients, Potential Anal., 2 (1993), 199-218. doi: 10.1007/BF01048505. Google Scholar
[32]	S. Littig and F. Schuricht, Convergence of the eigenvalues of the $p$-Laplace operator as $p$ goes to 1, Calc. Var. Partial Differential Equations, 49 (2014), 707-727. doi: 10.1007/s00526-013-0597-5. Google Scholar
[33]	M. Loss and C. Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal., 259 (2010), 1369-1379. doi: 10.1016/j.jfa.2010.05.001. Google Scholar
[34]	V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238. doi: 10.1006/jfan.2002.3955. Google Scholar
[35]	E. Parini, Continuity of the variational eigenvalues of the $p$-Laplacian with respect to $p$, Bull. Aust. Math. Soc., 83 (2011), 376-381. doi: 10.1017/S000497271100205X. Google Scholar
[36]	A. 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
[37]	Y. Sire and E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Commun. Partial Differ. Equations, 34 (2009), 765-784. doi: 10.1080/03605300902892402. Google Scholar
[38]	M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, forth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. Google Scholar

show all references

References:

[1]	L. Ambrosio, G. De Philippis and L. Martinazzi, $\Gamma$-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403. doi: 10.1007/s00229-010-0399-4. Google Scholar
[2]	D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. Google Scholar
[3]	J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan's 60th Birthday (eds. J. L. Menaldi, E. Rofman and A. Sulem), IOS Press, Amsterdam, 2001, 439-455. Google Scholar
[4]	J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s \to 1$ and applications, J. Anal. Math., 87 (2002), 77-101. doi: 10.1007/BF02868470. Google Scholar
[5]	H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983. Google Scholar
[6]	L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458. doi: 10.4171/IFB/325. Google Scholar
[7]	L. Brasco and E. Parini, The second eigenvalue of the fractional $p$-Laplacian, to appear on Adv. Calc. Var. (2015) doi: 10.1515/acv-2015-0007. Google Scholar
[8]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar
[9]	L. A. Caffarelli, Nonlocal equations, drifts and games, Nonlinear partial differential equations, Abel Symposia, 7 (2012), 37-52. doi: 10.1007/978-3-642-25361-4_3. Google Scholar
[10]	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
[11]	M. Cuesta, D. G. De Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian, J. Differential Equations, 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645. Google Scholar
[12]	G. Dal Maso, An Introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8. Google Scholar
[13]	M. Degiovanni and M. Marzocchi, Limit of minimax values under $\Gamma$-convergence, Electron. J. Differential Equations, (2014), 19pp. Google Scholar
[14]	A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, to appear on Ann. Inst. H. Poincaré Anal. Non Linéaire (2015) doi: 10.1016/j.anihpc.2015.04.003. Google Scholar
[15]	B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588. Google Scholar
[16]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. Google Scholar
[17]	E. R. Fadell and P. H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal., 26 (1977), 48-67. doi: 10.1016/0022-1236(77)90015-5. Google Scholar
[18]	A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253. doi: 10.5186/aasfm.2015.4009. Google Scholar
[19]	I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics. Springer, New York, 2007. Google Scholar
[20]	I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in $L^1$, SIAM J. Math. Anal., 23 (1992), 1081-1098. doi: 10.1137/0523060. Google Scholar
[21]	G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. Google Scholar
[22]	E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557. Google Scholar
[23]	A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006. Google Scholar
[24]	Y. X. Huang, On the eigenvalue of the $p$-Laplacian with varying $p$, Proc. Amer. Math. Soc., 125 (1997), 3347-3354. doi: 10.1090/S0002-9939-97-03961-0. Google Scholar
[25]	A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, to appear on Adv. Calc. Var., (2015) doi: 10.1515/acv-2014-0024. Google Scholar
[26]	A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, to appear on Rev. Mat. Iberoam.,, available at , (). Google Scholar
[27]	A. Iannizzotto and M. Squassina, Weyl-type laws for fractional $p$-eigenvalue problems, Asymptot. Anal., 88 (2014), 233-245. Google Scholar
[28]	T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368. doi: 10.1007/s00220-015-2356-2. Google Scholar
[29]	T. Kuusi, G. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57-114. doi: 10.2140/apde.2015.8.57. Google Scholar
[30]	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
[31]	P. Lindqvist, On non-linear Rayleigh quotients, Potential Anal., 2 (1993), 199-218. doi: 10.1007/BF01048505. Google Scholar
[32]	S. Littig and F. Schuricht, Convergence of the eigenvalues of the $p$-Laplace operator as $p$ goes to 1, Calc. Var. Partial Differential Equations, 49 (2014), 707-727. doi: 10.1007/s00526-013-0597-5. Google Scholar
[33]	M. Loss and C. Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal., 259 (2010), 1369-1379. doi: 10.1016/j.jfa.2010.05.001. Google Scholar
[34]	V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238. doi: 10.1006/jfan.2002.3955. Google Scholar
[35]	E. Parini, Continuity of the variational eigenvalues of the $p$-Laplacian with respect to $p$, Bull. Aust. Math. Soc., 83 (2011), 376-381. doi: 10.1017/S000497271100205X. Google Scholar
[36]	A. 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
[37]	Y. Sire and E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Commun. Partial Differ. Equations, 34 (2009), 765-784. doi: 10.1080/03605300902892402. Google Scholar
[38]	M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, forth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. Google Scholar

[1]	Leandro M. Del Pezzo, Julio D. Rossi. Eigenvalues for a nonlocal pseudo $p-$Laplacian. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6737-6765. doi: 10.3934/dcds.2016093
[2]	Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675
[3]	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
[4]	Raúl Ferreira, Julio D. Rossi. Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1469-1478. doi: 10.3934/dcds.2015.35.1469
[5]	Xianling Fan, Yuanzhang Zhao, Guifang Huang. Existence of solutions for the $p-$Laplacian with crossing nonlinearity. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 1019-1024. doi: 10.3934/dcds.2002.8.1019
[6]	Sergiu Aizicovici, Nikolaos S. Papageorgiou, V. Staicu. The spectrum and an index formula for the Neumann $p-$Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 431-456. doi: 10.3934/dcds.2009.25.431
[7]	Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252
[8]	Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743
[9]	Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[10]	Ravi P. Agarwal, Kanishka Perera, Zhitao Zhang. On some nonlocal eigenvalue problems. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 707-714. doi: 10.3934/dcdss.2012.5.707
[11]	Maria Fărcăşeanu, Mihai Mihăilescu, Denisa Stancu-Dumitru. Perturbed fractional eigenvalue problems. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6243-6255. doi: 10.3934/dcds.2017270
[12]	Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[13]	Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[14]	Ogabi Chokri. On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1157-1178. doi: 10.3934/cpaa.2016.15.1157
[15]	Elhoussine Azroul, Abdelmoujib Benkirane, and Mohammed Shimi. On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3479-3495. doi: 10.3934/dcdss.2020425
[16]	Dimitri Mugnai, Kanishka Perera, Edoardo Proietti Lippi. A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021177
[17]	Guowei Dai, Ruyun Ma. Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 99-116. doi: 10.3934/dcds.2015.35.99
[18]	Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013
[19]	Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008
[20]	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

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

Stability of variational eigenvalues for the fractional $p-$Laplacian

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Stability of variational eigenvalues for the fractional $p-$Laplacian

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors