June  2021, 41(6): 2993-3020. doi: 10.3934/dcds.2020394

Study of fractional Poincaré inequalities on unbounded domains

1. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway

2. 

Universitat de Barcelona, Spain

3. 

Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016, India

* Corresponding author: Gyula Csató

Received  May 2020 Revised  September 2020 Published  June 2021 Early access  December 2020

Fund Project: The first author is supported by the ERCIM "Alain Bensoussan" Fellowship program, HRI postdoctoral fellowship grant and Toppforsk project WaNP grant no. 250070. The third author is supported by Inspire grant IFA14-MA43 and Matrix grant MTR/2019/000585.
The second author is member of BGSMath Barcelona, part of the Catalan, research group 2017 SGR 1392, is supported by the MINECO grants MTM2017-83499-P and MTM2017-84214-C2-1-P and by the María de Maeztu Grant MDM-2014-0445

The aim of this paper is to study (regional) fractional Poincaré type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results on regional fractional inequality are established depending on various conditions on domains and on the range of $ s \in (0,1) $. The best constant in both regional fractional and fractional Poincaré inequality is characterized for strip like domains $ (\omega \times \mathbb{R}^{n-1}) $, and the results obtained in this direction are analogous to those of the local case. This settles one of the natural questions raised by K. Yeressian in [Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89, (2014), no 1-2].

Citation: Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2993-3020. doi: 10.3934/dcds.2020394
References:
[1]

R. A. Adams and J. Fournier, Sobolev Spaces, Second Edition, Pure and Applied Mathematics (Amsterdam), Vol. 140 2003, Elsevier/Academic Press, xiv+305 pp.

[2]

V. Ambrosio, L. Freddi and R. Musina, Asymptotic analysis of the Dirichlet fractional Laplacian in domains becoming unbounded, J. Math. Anal. Appl., 485 (2020), 123845, 17 pp. doi: 10.1016/j.jmaa.2020.123845.

[3]

L. Brasco and A. Salort, A note on homogeneous Sobolev space of fractional order, Ann. Mat. Pura Appl. (4), 198 (2019), 1295-1330.  doi: 10.1007/s10231-018-0817-x.

[4]

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

[5]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Mathematical Journal, 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621.

[6]

H. Chen, The Dirichlet elliptic problem involving regional fractional Laplacian, J. Math. Physics, 59 (2018), 071504, 19 pp. doi: 10.1063/1.5046685.

[7]

M. ChipotA. Mojsic and P. Roy, On some variational problems set on domains tending to infinity, Discrete Contin. Dyn. Syst., 36 (2016), 3603-3621.  doi: 10.3934/dcds.2016.36.3603.

[8]

M. ChipotP. Roy and I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity, Asymptot. Anal., 85 (2013), 199-227.  doi: 10.3233/ASY-131182.

[9]

M. Chipot and K. Yeressian, On the asymptotic behavior of variational inequalities set in cylinders, Discrete Contin. Dyn. Syst., 33 (2013), 4875-4890.  doi: 10.3934/dcds.2013.33.4875.

[10]

I. Chowdhury and P. Roy, On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity, Commun. Contemp. Math., 19 (2017), 21 pp. doi: 10.1142/S0219199716500358.

[11]

I. Chowdhury and P. Roy, Fractional Poincaré inequality for unbounded domains with finite ball condition: A Counter Example, arXiv: 2001.04441 (2020).

[12]

E. CintiJ. Serra and E. Valdinoci, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces, J. Differential Geom., 112 (2019), 447-504.  doi: 10.4310/jdg/1563242471.

[13]

B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588.  doi: 10.1215/ijm/1258138400.

[14]

B. Dyda and R. L. Frank, Fractional Hardy–Sobolev–Maz'ya inequality for domains, Studia Math., 208 (2012), 151-166.  doi: 10.4064/sm208-2-3.

[15]

B. DydaJ. Lehrbäck and A. V. Vähäkangas, Fractional Hardy-Sobolev type inequalities for half spaces and John domains, Proc. Amer. Math. Soc., 146 (2018), 3393-3402.  doi: 10.1090/proc/14051.

[16]

B. DydaL. Ihnatsyeva and A. Vähäkangas, On improved fractional Sobolev-Poincaré inequalities, Ark. Mat., 54 (2016), 437-454.  doi: 10.1007/s11512-015-0227-x.

[17]

L. Esposito, P. Roy and F. Sk, On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded, Asymptot. Anal., (2020), 1–16. doi: 10.3233/ASY-201626.

[18]

M. FelsingerM. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Mathematische Zeitschrift, 279 (2015), 779-809.  doi: 10.1007/s00209-014-1394-3.

[19]

A. FiscellaR. 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.

[20]

R. L. Frank, Eigenvalue bounds for the fractional Laplacian: A review,, Recent Developments in Nonlocal Theory, 210–235, De Gruyter, Berlin, 2018. doi: 10.1515/9783110571561-007.

[21]

R. L. Frank, T. Jin and J. Xiong, Minimizers for the fractional Sobolev inequality on domains, Calc. Var. Partial Differential Equations, 57 (2018), Art. 43, 31 pp. doi: 10.1007/s00526-018-1304-3.

[22]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 225 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.

[23]

R. Hurri-Syrjanen and A. V. Vähäkangas, Fractional Sobolev-Poincaré and fractional Hardy inequalities in unbounded John domains, Mathematika, 61 (2015), 385-401.  doi: 10.1112/S0025579314000230.

[24]

R. Hurri-Syrjanen and A. V. Vähäkangas, On fractional Poincaré inequalities, J. Anal. Math., 120 (2013), 85-104.  doi: 10.1007/s11854-013-0015-0.

[25]

D. Li and K. Wang, Symmetric radial decreasing rearrangement can increase the fractional Gagliardo norm in domains, Commun. Contemp. Math., 21 (2019), 1850059, 9 pp. doi: 10.1142/S0219199718500591.

[26]

J.-L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and Applications, Springer, Volume 1, 1972.

[27]

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.

[28]

G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068.  doi: 10.1142/S0219199710004111.

[29]

V. G. Maz'ja, Sobolev Spaces, Springer Ser. Soviet Math., Springer, Berlin, (1985). doi: 10.1007/978-3-662-09922-3.

[30]

C. MouhotE. Russ and Y. Sire, Fractional Poincaré inequalities for general measures, J. Math. Pures Appl., 95 (2011), 72-84.  doi: 10.1016/j.matpur.2010.10.003.

[31]

E. Di NezzaG. 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.

[32]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01.

[33]

X. Ros-Oton and J. Serra, Regularity theory for general stable operators, J. Diff. Equations, 260 (2016), 8675-8715.  doi: 10.1016/j.jde.2016.02.033.

[34]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.

[35]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.

[36]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matemàtiques, 58 (2014), 133-154.  doi: 10.5565/PUBLMAT_58114_06.

[37]

R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[38]

K. Yeressian, Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal., 89 (2014), 21-35.  doi: 10.3233/ASY-141224.

show all references

References:
[1]

R. A. Adams and J. Fournier, Sobolev Spaces, Second Edition, Pure and Applied Mathematics (Amsterdam), Vol. 140 2003, Elsevier/Academic Press, xiv+305 pp.

[2]

V. Ambrosio, L. Freddi and R. Musina, Asymptotic analysis of the Dirichlet fractional Laplacian in domains becoming unbounded, J. Math. Anal. Appl., 485 (2020), 123845, 17 pp. doi: 10.1016/j.jmaa.2020.123845.

[3]

L. Brasco and A. Salort, A note on homogeneous Sobolev space of fractional order, Ann. Mat. Pura Appl. (4), 198 (2019), 1295-1330.  doi: 10.1007/s10231-018-0817-x.

[4]

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

[5]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Mathematical Journal, 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621.

[6]

H. Chen, The Dirichlet elliptic problem involving regional fractional Laplacian, J. Math. Physics, 59 (2018), 071504, 19 pp. doi: 10.1063/1.5046685.

[7]

M. ChipotA. Mojsic and P. Roy, On some variational problems set on domains tending to infinity, Discrete Contin. Dyn. Syst., 36 (2016), 3603-3621.  doi: 10.3934/dcds.2016.36.3603.

[8]

M. ChipotP. Roy and I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity, Asymptot. Anal., 85 (2013), 199-227.  doi: 10.3233/ASY-131182.

[9]

M. Chipot and K. Yeressian, On the asymptotic behavior of variational inequalities set in cylinders, Discrete Contin. Dyn. Syst., 33 (2013), 4875-4890.  doi: 10.3934/dcds.2013.33.4875.

[10]

I. Chowdhury and P. Roy, On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity, Commun. Contemp. Math., 19 (2017), 21 pp. doi: 10.1142/S0219199716500358.

[11]

I. Chowdhury and P. Roy, Fractional Poincaré inequality for unbounded domains with finite ball condition: A Counter Example, arXiv: 2001.04441 (2020).

[12]

E. CintiJ. Serra and E. Valdinoci, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces, J. Differential Geom., 112 (2019), 447-504.  doi: 10.4310/jdg/1563242471.

[13]

B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588.  doi: 10.1215/ijm/1258138400.

[14]

B. Dyda and R. L. Frank, Fractional Hardy–Sobolev–Maz'ya inequality for domains, Studia Math., 208 (2012), 151-166.  doi: 10.4064/sm208-2-3.

[15]

B. DydaJ. Lehrbäck and A. V. Vähäkangas, Fractional Hardy-Sobolev type inequalities for half spaces and John domains, Proc. Amer. Math. Soc., 146 (2018), 3393-3402.  doi: 10.1090/proc/14051.

[16]

B. DydaL. Ihnatsyeva and A. Vähäkangas, On improved fractional Sobolev-Poincaré inequalities, Ark. Mat., 54 (2016), 437-454.  doi: 10.1007/s11512-015-0227-x.

[17]

L. Esposito, P. Roy and F. Sk, On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded, Asymptot. Anal., (2020), 1–16. doi: 10.3233/ASY-201626.

[18]

M. FelsingerM. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Mathematische Zeitschrift, 279 (2015), 779-809.  doi: 10.1007/s00209-014-1394-3.

[19]

A. FiscellaR. 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.

[20]

R. L. Frank, Eigenvalue bounds for the fractional Laplacian: A review,, Recent Developments in Nonlocal Theory, 210–235, De Gruyter, Berlin, 2018. doi: 10.1515/9783110571561-007.

[21]

R. L. Frank, T. Jin and J. Xiong, Minimizers for the fractional Sobolev inequality on domains, Calc. Var. Partial Differential Equations, 57 (2018), Art. 43, 31 pp. doi: 10.1007/s00526-018-1304-3.

[22]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 225 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.

[23]

R. Hurri-Syrjanen and A. V. Vähäkangas, Fractional Sobolev-Poincaré and fractional Hardy inequalities in unbounded John domains, Mathematika, 61 (2015), 385-401.  doi: 10.1112/S0025579314000230.

[24]

R. Hurri-Syrjanen and A. V. Vähäkangas, On fractional Poincaré inequalities, J. Anal. Math., 120 (2013), 85-104.  doi: 10.1007/s11854-013-0015-0.

[25]

D. Li and K. Wang, Symmetric radial decreasing rearrangement can increase the fractional Gagliardo norm in domains, Commun. Contemp. Math., 21 (2019), 1850059, 9 pp. doi: 10.1142/S0219199718500591.

[26]

J.-L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and Applications, Springer, Volume 1, 1972.

[27]

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.

[28]

G. Mancini and K. Sandeep, Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math., 12 (2010), 1055-1068.  doi: 10.1142/S0219199710004111.

[29]

V. G. Maz'ja, Sobolev Spaces, Springer Ser. Soviet Math., Springer, Berlin, (1985). doi: 10.1007/978-3-662-09922-3.

[30]

C. MouhotE. Russ and Y. Sire, Fractional Poincaré inequalities for general measures, J. Math. Pures Appl., 95 (2011), 72-84.  doi: 10.1016/j.matpur.2010.10.003.

[31]

E. Di NezzaG. 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.

[32]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01.

[33]

X. Ros-Oton and J. Serra, Regularity theory for general stable operators, J. Diff. Equations, 260 (2016), 8675-8715.  doi: 10.1016/j.jde.2016.02.033.

[34]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.

[35]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.

[36]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matemàtiques, 58 (2014), 133-154.  doi: 10.5565/PUBLMAT_58114_06.

[37]

R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[38]

K. Yeressian, Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal., 89 (2014), 21-35.  doi: 10.3233/ASY-141224.

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