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

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

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

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.
Citation: Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete and 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.

[2]

D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.

[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.

[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.

[5]

H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983.

[6]

L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458. doi: 10.4171/IFB/325.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

M. Degiovanni and M. Marzocchi, Limit of minimax values under $\Gamma$-convergence, Electron. J. Differential Equations, (2014), 19pp.

[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.

[15]

B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588.

[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.

[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.

[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.

[19]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics. Springer, New York, 2007.

[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.

[21]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386.

[22]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.

[23]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006.

[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.

[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.

[26]

A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, to appear on Rev. Mat. Iberoam.,, available at , (). 

[27]

A. Iannizzotto and M. Squassina, Weyl-type laws for fractional $p$-eigenvalue problems, Asymptot. Anal., 88 (2014), 233-245.

[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.

[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.

[30]

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

[31]

P. Lindqvist, On non-linear Rayleigh quotients, Potential Anal., 2 (1993), 199-218. doi: 10.1007/BF01048505.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.

[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.

[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.

[5]

H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983.

[6]

L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458. doi: 10.4171/IFB/325.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

M. Degiovanni and M. Marzocchi, Limit of minimax values under $\Gamma$-convergence, Electron. J. Differential Equations, (2014), 19pp.

[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.

[15]

B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588.

[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.

[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.

[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.

[19]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics. Springer, New York, 2007.

[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.

[21]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386.

[22]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.

[23]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006.

[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.

[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.

[26]

A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, to appear on Rev. Mat. Iberoam.,, available at , (). 

[27]

A. Iannizzotto and M. Squassina, Weyl-type laws for fractional $p$-eigenvalue problems, Asymptot. Anal., 88 (2014), 233-245.

[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.

[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.

[30]

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

[31]

P. Lindqvist, On non-linear Rayleigh quotients, Potential Anal., 2 (1993), 199-218. doi: 10.1007/BF01048505.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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